I heard from someone that the following problem is an open question.

(Open Problem 1)For a countable discrete group $G$, suppose it does not contain any Baumslag-Solitar subgroups $BS(m,n):=\langle x,y|xy^mx^{-1}=y^n\rangle$ and it admits a finite $K(G,1)$, is it a hyperbolic group?

I could not find the relevent stuff on this problem, so I am wondering whether the following is known.

(**My question**)For a countable discrete amenable group $G$, suppose it does not contain any Baumslag-Solitar subgroups $BS(m,n):=\langle x,y|xy^mx^{-1}=y^n\rangle$ and it admits a finite $K(G,1)$, is it a virtually cyclic group?

Note that ``Yes for problem 1 implies Yes for my question" since $G$ is amenable and hyperbolic iff $G$ is virtually cyclic(?)

Any references or comments are welcome!