[part 1/2]

> I wonder what happens if ... mond is used for galaxies that do the lensing

tl;dr: massive galaxy cluster with multiple images of a cosmologically-distant background supernova is a relativistic system, so neither Newtonian limit of GR nor MOND is suitable and thus defy useful comparison. Instead compare relativistic theories that reliably produce Newtonian limit (e.g. General Relativity + cold dark matter) or MOND (long list of relativistic theories...) at the limbs of galaxies.

The article at the top corresponds with the arXiv preprint <https://arxiv.org/abs/2305.06367>.

The foreground is a MAssive Cluster Survey (MACS, <https://iopscience.iop.org/article/10.1086/518603>) MACS J1149+2223, a large group of very X-ray luminous galaxies redshift z ~ 0.543 (about 5 billion light-years distance). As is typical for massive clusters, there are many multiple images of background events, concentrated around the centre of the cluster.

> mond

First, for convenience, let's just accept the empirical MOND relationship for all galaxies (i.e., we don't look at possible/apparent exceptions). Blobs of hydrogen gas at the tips of edge-on spiral-type galaxies have too large a red/blue shift for Newton+visible matter alone; and blobs of hydrogen gas scattered across the face of elliptical-type galaxies have too many outlying red/blue shifts (they move radially, rising through and sinking back below the bright surface). These features, wrapped up in "rotation curve", are compatible with a modified Newton (+ visible matter), where the modification in this limit is a simple function.

However, the combination of ~optical background sources and the centre of massive cluster demands a relativistic theory (see [1], which goes to your question in detail), and there have been many attempts to produce a relativistic MOND. Most of these involve adding an additional mathematical structure (e.g. a vector or scalar field) or more to the Einstein Field Equations to which all matter couples, driving these new gravitational fields' potentials, gradients, and dynamics. The goal, very broadly, is to have all the relevant gravitational dynamics produced only by the matter known from the Standard Model, interacting only with the mechanisms allowed in that model of matter. In these theories, the empirical MOND relationship is supposed to be inevitable in the weak-field / nonrelativistic limit.

By comparison, General Relativity adds a new species of matter to the Einstein Field Equations. The existing curvature tensor (and its decomposition fraction, the metric tensor) is the sole universally-coupled field. Hypothetical new matter (e.g. axions, sterile neutrinos, ...) would drive the metric tensor just like old matter; such new matter may or may not interact with other matter in non-gravitational ways. Unmodified Newtonian dynamics is inevitable in the weak field / nonrelativistic limit in the absence of this new matter. The empirical MOND relationship appears only if extra matter is added. The problem is that we don't see that extra matter, and do not know how (or if) it interacts non-gravitationally with the Standard Model.

(There are additional ideas, usually based in string theory/M-theory, wherein a relativistic theory of gravitation -- General Relativity, or one of the many flavours of relativistic MOND -- emerges at relatively low energies.)

Quoting [1] (§2),

The MOND paradigm has always struggled with rectifying the mass
discrepancy in galaxy clusters without including an extra mass
component such as neutrinos or non-luminous baryonic matter. In its
usual form MOND is unable to boost the gravitational
acceleration in clusters sufficiently whilst this framework is
still able to match observational constraints provided by galaxy
dynamics. The reason for MOND’s inability to accomplish this is that
the gravitational acceleration in galaxy clusters is relatively high,
in general x>1, so the MOND boost to gravity is weak.
However, there is an observed mass deficit in galaxy clusters that
would require the cluster to be in the deep-MOND regime (x<<1) to
rectify with MOND alone.

Put more simply, the mass profile of the centre of massive clusters makes constituent galaxies whirl around within the cluster too much for MOND alone. There are a variety of ideas about how this could be fixed, mostly by building a more complicated relativistic theory for which MOND (modified Newton + no "new" matter) continues to be the weak-field limit. The General Relativity answer is that the (luminous-)mass deficit implies the presence of dark matter in regions of the cluster far outside any of its galaxies.

[continued in part 2/2]