Question

Well, it does according to this preprint for certain scales.

What would be a simple way to explain MOND to a layman?

Does it ignore mainstream physics? How much?

Answer 1

Yes. The would appear to be the point of the paper you cite. It has been accepted for PRL apparently so that should lend it some credibility.

MOND - short for MOdified Newtonian Dynamics - is a phenomenological theory that was conceived of by Mordechai Milgrom in order to explain the huge discrepancy between the shapes of galaxy rotation curves as predicted by Newtonian theory and the actual shapes observed. A rotation curve is the plot of the orbital velocity of objects in the galaxy with respect to distance from the center:
The simplest statement of MOND is that it is a theory with a minimum acceleration scale . When this scale is reached at some radius in a galaxy, objects appear to cease to respond to gravitational forces. Alternatively one could say that gravitational forces cannot generate an acceleration lower than .

There are many critics of MOND. However, it has had remarkable success over the years. This paper is only the most recent evidence in its favor. MOND makes no claim to explaining the microscopic physics which leads to these effects. Compared to the LCDM model, MOND makes far fewer assumptions (only one, in fact) and thus has the benefit of being simpler.

The challenge for theories of quantum gravity is to either rule out MOND'ian behavior in weak fields or otherwise explain what gives rise to it. So far, none have even tackled the question in part due out of fear of being labeled "fringe" for associating with such rabble as MOND ;]