It looks moderately muddled, but perhaps. This paper1 is a good place to start; it's free and the intro lists a number of more recent studies. That study found a connection to lifetime reproductive success in sticklebacks (a fish) with intermediate MHC diversity. This 2007 study in mice2 is a particularly nice one, dealing with a inbred line outbred to wild mice, finding again, no advantage to heterozygotes. There are others, but most deal with the concept descriptively (self-delighting in their continued refutation of a commonly-held belief) or predictively (such as this from PNAS in 19923 or a 2009 review4) but specific and proven reasons are sparse.
The general feeling is that it may remain unclear what sort of genetic interactions are involved in MHC (additive effects, overdominance, etc.) but it is probably also complicated due to the nature of immunology. For example, a paper from 20035 points out that in an immunological response, only a few peptides dominate the response; there may be no real advantage to having extra ones if they are unused, and having those others around might dilute the useful response (That paper also takes an interesting view of serial infections, and shows that heterozygotes are indeed advantaged during coinfection). The above PNAS publication3 gives the idea, which the stickleback paper1 recapitulated, that if more alleles are present, more T-cells will be self-reactive and will therefore be purged, resulting in an overall decreased T-cell availability.
Other hypotheses are present in the review4 I mentioned, such as perhaps positive selection efficiency is not MHC-diversity dependent whereas negative selection is, or indeed increased autoimmune reactions as you guessed. My guess would be some combination of the above, with selection for heterozygotes being favored in the real world for multiple and repeated infections but certain specifics resulting in a homozygote preference, of which autoimmunity is likely playing a role.