That brings up the question of what one means by "sex". As I understand it, sex is primarily about the kinds of gametes one produces - either large gametes (female) or small gametes (male). The genetic implementation of that is in some sense a secondary detail, since it varies between organisms. Reptiles, for instance, only have XX chromosomes and sex is determined by environment (or something like that).

I would expect that in this case the mushroom still only has two types of gametes, and hence only two sexes. There's just an added bit of complexity determining whether a particular pair of gametes can produce a new organism.