Perhaps I am missing something here but it seems quite intuitive to me that it has to be 50-50.
Think of it as coin tosses: As the resulting distribution is completely independent of the stopping time the proportion will in the limit always converge to the original distribution which is supposed to be 50-50. That is because the resulting stochastic process is markovian and a martingale
Or put another way: It doesn't matter when you stop tossing, the outcome will always add up to 50-50 because the coin doesn't have a memory.

Does this make sense?

Perhaps I am missing something here but it seems quite intuitive to me that it has to be 50-50.
Think of it as coin tosses: As the resulting distribution is completely independent of the stopping time the proportion will in the limit always converge to the original distribution which is supposed to be 50-50. That is because the resulting stochastic process is markovian and a martingale
Or put another way: It doesn't matter when you stop tossing, the outcome will always add up to 50-50 because the coin doesn't have a memory.

It is a little bit like trying to invest in the stock market and getting out every time you are in the plus (sitting out negative phases) and after that beginning all over again. This seems like a clever infallible strategy - alas it doesn't work and you will stay at zero in the long run (here minus transaction costs of course)

(BTW: This reminds me of some friends of ours who desperately wanted a boy - now they have three girls and stopped "trying"... ;-)