I hope this helps. In order to get anywhere close to "25" I will ignore some factors and engage in a bit of "lie to students" terminological and notational abuse.

> ... may be helpful to discover an extension of the theory.

The authors have a set of families of alternatives to General Relativity, and are exploring whether they are viable as a physical description of gravitation in our universe, and if not (spoiler: they're not; they admit this up front) whether any of them can be adapted into a good physical description.

The second author of the (purely theoretical) headline paper in [(headline)paper ref [43]] explored variations on Brans-Dicke (BD) scalar-tensor gravitation.

Briefly, BD adds an additional scalar field to the familiar Einstein curvature tensor of General Relativity. The field is physically interpretable as a gravitational potential, or more precisely, a value that takes the global value of the gravitational constant G to a local effective gravitational constant G_{eff}. One as a result has to adapt the relationship between the curvature and the distribution of stress-energy, by introducing a coupling between matter and the scalar field. In BD, this is the Brans-Dicke coupling constant \omega. The higher the value of \omega, the more matter couples to the tensor part and the less to the scalar part of the scalar-tensor gravity. The value of \omega is not constrained by the theory: one has to look at the behaviour of the distribution of matter to constrain \omega.

In General Relativity, matter at a point can completely determine the Einstein curvature tensor at that point. In B-D, however, the scalar potential also has a role. So if the matter sources a solution in General Relativity, one needs an appropriate \omega to source a solution in B-D for a given value of the scalar potential.

The more we look at astrophysical phenomena, and the more we do direct tests in our solar system, the more strongly positive the lowest physically compatible value of \omega can be, or the closer to zero the magnitude of the scalar must be, or both.

If the value of the scalar is everywhere zero, it can simply be elided, and one gets General Relativity.

At \omega -> \infty BD has for all practical purposes identical behaviour to General Relativity for any value of the scalar. Parsimony in that case would favour GR: the scalar field, even at its |strongest|, would just not affect matter.

A goal of theoreticians trying to keep scalar-tensor theory alive and relevant as a physical possibility therefore is to have \omega very high everywhere except near compact objects such as (stellar!) black holes and neutron stars. Around them, they might hope, the strength of the self-interaction of the scalar field (like the self-interaction of the Einstein curvature tensor in General Relativity, and the tensor part of scalar-tensor) leads to interesting differences from General Relativity.

The result is a "cloud" of scalar self-interaction around compact objects caused by the very high but still not infinite coupling constant, and the presence of matter. This is called "spontaneous scalarization". In [paper ref [43]] the headline paper's second author explores whether there might be "spontaneous vectorization" if one substitutes a vector field for the scalar field. It does, and then generalizes to spontaneous "self-gravitating gravitational cloud" formation in any theory where matter couples to an additional gravitational component beyond the Einstein tensor.

Boiling this down: 2nd author Ramazanoğlu has been keen on finding some extra knobs to twiddle in strong gravity while ensuring the knob-twiddling produces no measurable effect in weak gravity, and he has a preference for vector-tensor rather than scalar-tensor gravity. In these, it's a dynamical vector field that couples to matter (only) gravitationally, so again one has a coupling constant to choose. Although the details of parametrizing the coupling differ from Brans-Dicke, the coupling must be weak in a universe with the sorts of structures that exist in our sky.

As he and his co-authors write in the headline paper, they are excited by the theoretical possbility that there may be '... new fundamental fields [which] lie "dormant" in weak-gravity environments and yet manage to have significant effects in strongly gravitating bodies and systems'.

Which is perfectly reasonable for a theorist, driven by theoretical considerations.

There is no astrophysical justification for such fields, however (and some evidence against them) and generically the effects of these fields have to die away around ordinary stars and large black holes, or collapsing systems (star and planet formation, galaxy and galaxy-cluster evolution) don't match what we see in the sky.

Worse, there are some further problems for the theorists: [a] there is evidence in multi-pulsar systems (Hulse-Taylor double and PSR J0337+1715 triple) for the strong equivalence principle (SEP), and one cannot faithfully reproduce the SEP when the magnitude of the vector component is large, as it must be near compact objects (like pulsars); [b] gradient instabilities where there are modes with negative propagation speed squared lead to an explosion of self-gravitating-gravity that is nothing like what we see in our universe, and is almost certainly incompatible with the hot big bang; [c] the presence of quantum ghosts, leading to an unstable vacuum state.

The [c] ghosts are the topic of the headline paper.

Start with a classical vacuum vector-tensor background, then equip it with a (matter) quantum field where we freeze every point in the quantum field at the vacuum expectation value. Now add a compact mass, and see what happens to the vector-scalar gravitational fields, and the matter quantum field as they interact according to the coupling. Do we get negative energy modes? If so, we must have https://en.wikipedia.org/wiki/Ghost_(physics)#Bad_ghosts whose causal cones are much wider than those of massless particles. Mechanisms that suppress negative energy modes break Lorentz invariance, and we have very very very good evidence that we have Lorentz invariance everywhere in our sky.

One might, with enormous care, contrive a globally hyperbolic (and thus strongly causal) toy universe even with these FTL ghosts, following some logic proposed by Bob Geroch (https://arxiv.org/abs/1005.1614). Geroch's argument is that one could not say a priori that such a toy universe would -- or even could -- develop structures like the ones in our universe. One would have to just go compute the evolution and find out.

At least two of the authors of the headline paper specialize in computational gravitational physics, so maybe that's on their todo list. (cf. their Acknowledgments section, final sentence).

In this paper they only went as far as finding out that (a) being haunted by bad ghosts seems pretty generic where one has a novel auxiliary gravitational field that is strong only in strong gravity and (b) there may be refuge in adapting the non-linear self-interactions. As a slogan, "gravity self-gravitates, but the vector and tensor parts self-interact differently". It's an interesting theoretical idea, but there won't be astrophysical evidence to support or refute it for at least many years, and only then if they commit to choosing a range of couplings (rather than adjusting the coupling constant to match new evidence if and as it appears, which would be awkward if the new evidence does not force them into closer accord with General Relativity, as has happened to Brans-Dicke). Indeed, this entire programme of "spontaneous scalarization" and "spontaneous vectorization" is because there is so much observational (and in-solar-system experimental) support for General Relativity, meaning they have to hide their differences from GR in less and less observationally-accessible settings. In particular, they explicitly aim for complete compatibility with General Relativity in the weak field limit (give or take some thoughts on gravitational waves), and differences only close to small compact objects, or deep within large black holes (and thus hidden from practical observation).