HLA's are not specifically related to mold in general. Their primary use in medicine is in histocompatibility - that is, in transplantation from one person to another. They determine whether 2 people are compatible enough to swap organs. No 2 people are perfectly histocompatible except for identical twins, although 1 in 4 siblings will be very close. My sister and I are not identical - we match on one chromosome, but not the other. So for kidney or liver transplantation, that would mean we are a 3/6 match, which is usable but requires more immunosuppression than a 6/6 match. (Generally, you'll be 0/6, 3/6, or 6/6 with a sibling since they are closely linked and transmitted together). For bone marrow transplantation, we would be 5/10 - which isn't really very good. A good match is far more important for bone marrow, and HLA-DR/DQ and to a lesser extent DP can matter.
The other very interesting aspect of HLA's, and closer to why I ask here, is that they are major determinants of immune function, and hence, susceptibility or resistance to autoimmune/autoinflammatory disease and infectious disease. The strongest association is probably HLA-B27, which is very strongly associated with ankylosing spondylitis, psoriatic arthritis, inflammatory bowel disease, reactive arthritis, and similar diseases. HLA-DR4 predisposes to rheumatoid arthritis, as do some subtypes of HLA-DR1 (not DRB1*01:03, but the more common DRB1*01:01). HLA-DR11 predisposes to systemic sclerosis. HLA-A1 is protective against type 1 diabetes. HLA-DR3 predisposes to multiple sclerosis. There are many other examples, and they affect autoimmune diseases as well (e.g. B27 carriers are relatively resistant to HIV, but B35 are susceptible - with Lyme, DR3 carriers are susceptible to chronic lyme whereas DR11 are resistant - that's probably more autoimmune than infectious, although it may relate to how rapidly and effectively the bacterium is cleared).
HLA's may also affect allergies, since the antigen fragments that are presented (and hence, against which the body forms antibodies - including IgE, which is involved in allergies) are determined by the HLA's. It's possible there is a connection to mold allergy.
Personally I think Shoemaker has some interesting observations, but I think his interpretation is completely wrong. I don't think all these patients are exposed to mold or other biotoxins. Maybe a few are - if any... From what I hear though he was really obnoxious to his patients and is one of those docs who really wants to make a lot of money. I have no idea if the cholestyramine works at all - I have not tried it and have no plans to. My illness has not been better or worse living in multiple states and multiple places in each state - so I really don't buy the mold idea. He's not really able to prove the mold connection, although he has shown some abnormalities in patients, mostly autoimmune/endocrine/neuro.
I wouldn't put him with Dr. Goldstein or Dr. Myhill. I know Dr. Goldstein was incredibly dedicated to his patients, and never made any money doing what he did. He ended up in trouble with the California medical board only because they couldn't understand his work. Also, a lot of his work was trial and error. I think many of us would appreciate that though - a doc who will go through 500 meds until we feel better. Most of us are lucky to try a new drug a few times a year, and most of the time they do not work, and many of the drugs that Goldstein found most effective are not drugs the average doc would use. Goldstein stepped up and helped his patients because no one else would/could. I don't know Myhill as well but I think her case is similar. For example her website is free and contains pretty much all of her thinking on the subject. Not true of all docs. That doesn't mean she's right - I don't know - but I think she is trying and thinking about it, and she's doing it to help her patients. She's also unfortunately in the UK (unfortunate for her) - the UK has one of the most backwards medical systems right now in treating patients with ME thanks to Wesseley and the psychobabble crowd having gained so much power because their treatment approach is cheapest...
Norway seems to be one of the most progressive. Not sure about how Belgium is, but I know DeMeirleir is there - but he may be a speciality physician.
@Valentijn has said generally the Dutch system isn't very good for ME patients at all. Canada isn't so great either based on Dr. Hyde's descriptions. The UK is probably the worst though because of Wesseley's influence and huge budget cuts since the time of Thatcher that have adversely affected the quality of care for patients, especially those with strange illnesses like ME.