That's like asking "If homeopathy worked and all the doctors were wrong, would they admit it?" You can't just flip a bit in the world setting Homeopathy_Works to TRUE and keep everything else the same.
You can look at cases like chiropractors. Over a long time there was a general belief that chiropractors didn't provide any good for patients because they theory based on which chiropractors practice is in substantial conflict with the theories used by Western medicine.
Suddenly in 2008 Cochrane comes out with the claim that chiropractors actually do provide comparable health benefits for patients with back pain as conventional treatment for backpain.
A lot of the opposition to homeopathy is based on the fact that the theory base of homeopathy is in conflict with standard Western knowledge about how things are supposed to work.
People often fail to notice things for bad reasons.
There are very good reasons why finding that one set of studies shows an unusual result is not taken as proof by either doctors or scientists. (It is also routine for pseudoscientists to latch onto that one or few studies when they happen.)
In other words, chiropractic is not such a case.
[the] theory based on which chiropractors practice is in substantial conflict with the theories used by Western medicine.
I hope you're not suggesting that the theories used by Western medicine are likely to be wrong here.
I just finished the first draft of my essay, "Are Sunk Costs Fallacies?"; there is still material I need to go through, but the bulk of the material is now there. The formatting is too gnarly to post here, so I ask everyone's forgiveness in clicking through.
To summarize:
(If any of that seems unlikely or absurd to you, click through. I've worked very hard to provide multiple citations where possible, and fulltext for practically everything.)
I started this a while ago; but Luke/SIAI paid for much of the work, and that motivation plus academic library access made this essay more comprehensive than it would have been and finished months in advance.