The issue is really how one thinks about "contrast". There's microcontrast which is intrinsically linked to both sharpness and chromatic aberration, colour contrast, and what people talk about as global contrast. The latter of which is, I tend to think, the most commonly conjured in ones when someone says contrast, despite it being a little simplistic.
The first of these is clearly affected by lens coatings, but is mostly the result of the optical design and how well the lens performs achromatically over the spectral (visible) range.
The second is not what I'd first think of when someone talks about contrast generally. This is to do with the spectral transmission function of the coatings, and the comparison between uncoated, single coated and multicoated. On a global (over the whole image) scale, this will affect the perceived 'contrast', as certain tones may suffer more of less loss (difference in the transmission percentage) than others, thereby affecting the contrast as compared with a different coating or no coating at all, and indeed the perceived relative brightness between that tone/colour and another. So this is really more the global contrast that people think of outside of contrast that you might alter with general levels in post. The issue is that there are multiple effects which are all amalgamated into the word contrast, some effects are independent, others are linked to one another, so it's really not a trivial thing to explain in a forum post.
Doing what I do, I tend to work from a more fundamental effect upwards. As such, when someone says contrast, I immediately simplify to monochromatic light and go from there. Say for example that you illuminate a piece of perfectly white paper with only a monochromatic light but such that there is a range of illuminating intensities with step-like boundaries (like a test exposure on a new dark room enlarger) - the brightness would be different on one part of the paper than another. If you photograph this, you could say there is range of contrast between the brightness regions. As the light is monochromatic, the lens coating, or lack thereof, will not alter the perceived contrast between the brightness regions when compared to the opposite lens coating case. In other words, if the lens offers 4% loss at that colour or 0.4% loss, and if the overall exposure is the same, the contrast will not change. This is physically impossible without a non-linear process. But, the sharpness at the boundaries between the brightness regions may still change.
Now, as soon as multiple colours are used, as in a real scene, now the global contrast may vary depending on the coatings. But it must be understood that this is a property of the spectral profile of the coatings and hence how different colours are transmitted more or less than compared with another coating or no coating at all. AND, (and this is where my issue mainly comes from) this does not imply that contrast will be stronger. It may well be that, given two colours in a scene, their relative brightness difference may be reduced if the coatings work that way. Therefore the notion of adding coatings always improves contrast is nonsense A) because of the above, and B) because "improve" is a relative term.