4 problems

Fibered homotopy K-theory isomorphism conjecture. The group $G$ satisfies the (fibered) $\mathit{KH}$-isomorphism conjecture for $R$ if the pair $(G,\mathcal{FIN})$ satisfies the (…

Family-stabiliser tree conjecture. The inequality $\operatorname{cd}_{\mathcal{F}}G\leqslant 1$ holds if and only if $G$ acts on a tree whose stabilisers generate $\mathcal{F}$.

Type I conjecture. Under these assumptions, $G$ is a type I group.

Open amalgams conjecture. Every completely metrisable topological group has property (OA).