I recently looked into the Alpha Centauri system in preparation for a talk I went to see on it. This system is also called Rigil Kent, a great name for a superhero, and is the closest stellar system to us (only 4.37 light years away) with two stars, A and B, similar to the Sun which form a tight binary system orbiting every 79 years, and a third called Proxima Centauri which is much further out and far smaller in size. The interesting thing for me is that little Proxima is so far out (13,000 AUs) that its acceleration with respect to the other two is in the regime where MiHsC should apply (of order 10^-10 m/s^2).
MiHsC says that a body with such a low acceleration relative to nearby matter (stars A and B) will lose some of its inertial mass in a new way, and this means it will be more easily bent gravitationally into a bound orbit even by a lower than expected central mass. This is exactly how MiHsC predicts bound galaxy rotation without dark matter (McCulloch, 2012) and it also predicts that Proxima should be bound gravitationally to A and B even though their masses should appear to be too small to bind it.
From my limited reading of papers so far, this seems to be the case. Proxima moves through space with stars A and B so it looks like it's bound to them, but Matthews and Gilmore (1993) found that according to Newton's laws Proxima should not be bound. To fix this problem they suggested increasing the mass of A and B to hold Promixa in to the system. Of course, this sounds just like the dark matter fix used to allow galaxies to remain bound without changing Newton's laws. The great thing for me about this Centauri mismatch is that dark matter cannot be used to explain it, since dark matter has to stay spread out on these small scales if it hopes to explain the galaxy rotation problem.
To prove MiHsC I've been looking for a problem for which it is the only possible solution: a crucial experiment. I might have found one in our cosmic backyard.
Matthews and Gilmore, 1993. MNRAS, 261, L5
Wertheimer and Laughlin, 2006. Are Proxima and Alpha Centauri Gravitationally Bound? Astron. J., 132, 1995-1997. http://arxiv.org/abs/astro-ph/0607401