Freemasonry and the GA

I noted when doing some research on Presidents that several of them were members of the Freemasons, which was only open to men at the time, but now has parallel branches for women.

Sir Thomas Henry Holland (1938) and Sir Richard Arman Gregory (1924) were both members of the Imperial Lodge college of Freemasons.

I wondered to what degree membership of the Freemasons may have been a factor in selection to the GA Presidency, if at all. I guess this not something that we are going to find out, and it may not be relevant at all. I also think it's fair to assume that many organisations would have had Freemasons at this time.

I know very little about this, and it may be those were the only two Presidents who were Freemasons and they didn't even know each other, but I'm guessing there may have been others, given the high profile of many of the Presidents in the first third of the 20th Century in particular.

More information would be gratefully received if anyone knows anything further.

Update
Doing some image searches for various Presidents, I found that John Linton Myres became a Freemason in 1899 and also came across some images of badges and recognition of work that he had been awarded.