I'd add https://arxiv.org/abs/1603.03312 too, as a link to the actual paper's arxiv version.

Bimetric theories are pretty strongly disfavoured on the grounds that there is an apparent universal coupling to the single metric g_\alpha\beta which lets us treat the metric as a geometrical property of spacetime, which gives us General Relativity everywhere we can presently look, and even in places we can't (GR as an effective field theory should hold up inside the horizons of the largest black holes until fairly close to the centre of mass).

One can try for a UV completion of General Relativity in a variety of ways, and broken symmetry approaches are fairly popular. In essence, an energy-dependence in the speed of massless matter particles can be transformed into a time-dependence, and with some care one can arrive at equal speeds for massless matter waves and massless gravitational waves at some plausible point in the early universe, with a cutoff that forbids observation of time-dependent speed of light in the later universe (and in particular not in our laboratories or in emissions from inspiralling black holes).

Cosmic Inflation also relies upon symmetry breaking at the end of the inflationary epoch, but does so without changing the behaviour of the matter fields; forbidding observable patches of inflation in the modern universe is a requirement.

Partly consequentially, it seems perfectly reasonable to wonder whether a bimetric theory could decay to General Relativity early after the big bang. Rosen proposed a bimetric theory in 1940, for example, and it was only effectively ruled out by indirect observations of loss of momentum from binary pulsars (notably Hulse-Taylor), and has only just been killed off thanks to the results from LIGO. Others have proposed bimetric theories since Rosen, with varying ways of matching the observables that we have at present (since all the observables are consistent with General Relativity).

A much higher temperature for the phase transition pushes observable effects away from all but the highest-energy GW producers after galaxy formation, but increases gravitational wave artifacts in the relic fields like the CMB and its neutrino and possibly other equivalents. These artifacts are central to the testability claim of the authors. Magueijo elsewhere (the authors' endnote [7]) has argued that adding in modified dispersion relations also solves many of the issues cosmic inflation does (spectral index, horizon problem).

Peter Coles highlights his own comment here:

https://telescoper.wordpress.com/2016/11/28/nailing-cosmolog...

It's hard to improve upon that. Observational support for Afshordi & Magueijo's idea -- barring any theoretical problems they discover in further elucidating it -- will only constrain cosmic inflationary models, rather than kill them off. Conversely, observational evidence sufficient to kill off their idea will not kill off bimetric gravity theories, it can only constrain them.

(One could argue that either way it's good science, since constraining the space of viable theories is always useful.)

Finally, Teh Grauniad's article offers up this direct quote. I hope they got it wrong: 'David Marsh, of the Centre for Theoretical Cosmology at Cambridge University, is not giving up on inflation yet. “The predictions of inflation developed by Stephen Hawking and others...' It is a serious oversight not to lead off such a list with Alan Guth, even if he's not at Cambridge or even British. [ https://en.wikipedia.org/wiki/Alan_Guth ]

ETA: finally, finally, another way of considering this idea is that in General Relativity the geometry of spacetime induces the Poincaré invariance on the tangent spaces (by inducing the Minkowski metric on them). That is how we get the symmetries of Special Relativity (which are baked into the Standard Model) at every point. A bimetric theory cannot induce this symmetry everywhere, and so cannot be valid anywhere the Standard Model is. So bimetric approaches are almost obliged to decay so early that before the decay is unobservable even in principle. The paper here does the same, but with the idea that the relic fields will hold a signature of the decay. By implication, the relic fields should also hold a signature of Beyond-the-Standard-Model physics, and maybe ultra high energy astrophysical processes in nature ought to reveal such BTSM physics too.