TITLE: Are Hyperbolic Groups Residually Amenable
QUESTION [10 upvotes]: It is a well-known conjecture (or maybe just a question) that all hyperbolic groups are residually finite. What happens if we weaken the conclusion; in particular
Are all hyperbolic groups residually amenable?
What is known in this direction?

REPLY [10 votes]: Proposition 7 of this paper establishes that every hyperbolic group is residually amenable iff every hyperbolic group is residually finite.