I think they believe it. I am sure there are a few who know it is false, or know that is has some serious problems, but by and large I think they are TBMs
I think some of them probably know, and you can probably tell which ones they are. The ones that tend to sound like ass-hole politicians during their general conferences, I'm willing to bet, are the ones that know damned well they are in charge of, more or less, a cult. However, the ones that generally try to be good (often failing, but at least they're trying right?) are probably just those who have been pulled from the mindwashed ranks. The reason I'd be willing to believe that all of them know is because church members are taught to be obedient and have their lives structured for them. To suddenly give one the power to do the structuring, I doubt they'd be as efficient as they are.
I think most or all of them are so invested and convinced of its "truthiness" that they can't even allow themselves to question "what if?" etc. I think some are aware of some of the dirt in Mormon history, but they don't allow it to make them doubt.
I think that is the kicker, the LDS church has a tremendous amount of power and money connected with it as well as every aspect of these men's lives. To go against the grain of that would mean losing all that they have in so many ways.
How could they be in those high positions and not know about the constant efforts made to keep haters quiet and anyone who dares to question the Church? Is that very Christlike...not quite...they know but they can't do anything about it because they get way too invested in that whole world and probably almost forget about the real world out there. OR they live one way in the public eye and another way in their private lives.
1. They are aware of the "issues" because they give talks on the subjects (re: Holland) and they receive letters from people who have concerns about the "issues".
2. Therefore some of them have reached a conclusion it is false. They continue on for the same reasons some of us do.