HGIYLHE Question on Hand 4

I posted this over in SS, but maybe it should go here...

For those who have this book, perhaps you could comment on the river play in Hand 4. If it is recommended to check/call here, why would it not be better to bet/fold instead? Seems like we can be pretty sure the villain will bet if we check, and that we are likely beat...but either way it costs us 1 bet.

However, it seems to me like villain might fold even two pair (assuming he has AA88 with a better kicker) if we bet. We c/r on the flop and 3-bet the turn...he could be very well convinced we have the 8, no?