Are there any published stats on whether P&W, RR, or GE engines have been more reliable in the 777?
I haven't found any.But that is not to say it does not exist.
One will recall that GE90's had a pretty rough going the first year of introduction and BA (first GE powered 777 customer) had to withdraw its 777s for several months until problems were fixed. Kind of similar to the problems that PW is having with 320neos.
BTW, Beyond 777-200ER, and 777-300 (not ER) the only engine available is various versions of GE90/9X. That is 777-200LR/F, 777-300ER and 777X versions have only GE90 or GE9X variants. No RR or PW engines for them.