One of the complaints I often get is that MiHsC violates the equivalence principle, which says that the inertial and gravitational masses are equal. This principle forms the basis of general relativity (GR) and has been well tested, so you might say that to suggest a theory that disagrees with it takes some balls.
Speaking of balls, legend has it that Galileo first demonstrated equivalence by dropping two of them of different mass off the leaning tower of Pisa, and showed that they hit the ground together. This occurs because, although the heavier ball has more gravitational mass and is attracted by the Earth more, it also has more inertial mass, so finds it harder to accelerate. The cancellation of these two effects is the equivalence principle. These days people do the same experiment far more accurately using a torsion balance: a couple of balls of different masses at either end of a rod in a dumb-bell like arrangement. The rod is suspended from its mid point by a wire whose resistance to twist is known. The differential horizontal 'falling' of the two balls of different mass towards distant objects like the Sun or the galaxy, is determined from the twist in the wire. No violation of equivalence has been observed down to unfeasibly tiny accelerations.
The reason this does not disprove MiHsC is clear, but subtle. MiHsC predicts that inertial mass is caused by Unruh radiation and for normal accelerations such as those on Earth or in the inner Solar system it predicts the standard inertial mass. MiHsC starts to deviate from normal when accelerations become ridiculously small, for example for the slowly curving stars at the edges of galaxies, because then the Unruh wavelengths the stars see, and that MiHsC says cause their inertia, get extremely long and a lesser proportion of them fit exactly within the Hubble scale. As a result logic says they cannot be observed (we have to assume there's no space outside the Hubble sphere for them to wiggle in) and therefore they cannot exist. This means that MiHsC predicts that the stars' inertia reduces in a new way for low accelerations, and a new acceleration appears which is 'independent of the objects' mass' and of size: 2c^2/(Hubble scale). This is intriguingly the same size as the cosmic acceleration and also fixes the galaxy rotation problem.
The crucial point is that no matter at what tiny acceleration equivalence is tested using the balls, the effects of MiHsC will not be seen, because the extra MiHsC acceleration is independent of the mass, meaning that Galileo's two balls, or the balls in the torsion balance, will still 'fall' equally, so there will be no twist in the wire and equivalence will appear to be unbroken. The difference is that with MiHsC the two balls both fall very slightly faster. What this says about the relationship between MiHsC and general relativity (which was build on equivalence) I do not yet know.
Having told you how MiHsC cannot be seen, it's only fair to tell you how it could be seen. One method is discussed here. Another interesting method would be to take the MiHsC acceleration: a=2c^2/(Hubble scale), and reduce the Hubble scale from its usual value of about 10^27 m (as big as it gets, so it makes the MiHsC effect tiny), by instead creating a closer horizon using a metamaterial structure (as I suggested in the last section of the first reference below, please ignore the first part which has been superceded by a later paper). Such a structure, or cavity, would boost the new MiHsC acceleration. The emdrive may be doing this, since MiHsC predicts the observed thrust quite well (see the second reference below) and this blog entry.
McCulloch, M.E., 2008. Can the flyby anomalies be explained by a modification of inertia, JBIS, 61, 373-378. ArXiv link (please see only section 4!).
McCulloch, M.E., 2015. Can the emdrive be explained by quantised inertia? Progress in Physics, 11, 78-80. Link