Does the apostrophe in names of diseases drop when the disease name is part of a foundation name?

Many diseases are named after the doctor who identified the diseases and therefore have names like Hodgkin's Disease or Parkinson's Disease. But when a fund or foundation is created to raise money for research on the disease, should the apostrophe remain? In other words, should it be Parkinson's Foundation or Parkinsons Foundation? If the apostrophe is dropped, is it dropped out of necessity?

Most commonly, the apostrophe AND the letter "s" are omitted, as seen in the following organisations, among others:

• National Parkinson Foundation
• American Parkinson Disease Association
• Huntington Society of Canada

In some other cases, the apostrophe is used, as in "Northwest Parkinson's Foundation".

Generally, "Parkinsons Foundation" is not used.

The (US) National Down Syndrome Society has a 'Preferred Language Guide' on their web site that says in part:

Down vs. Down's - NDSS uses the preferred spelling, Down syndrome, rather than Down's syndrome ... This is because an "apostrophe s" connotes ownership or possession. Down syndrome is named for the English physician John Langdon Down, who characterized the condition, but did not have it.

While this may not apply to other diseases or syndromes, it's a useful starting place.