Kropholler's conjecture on splittings over subgroups of codimension-1 subgroups

Let $G$) be a finitely generated group and let $H$ be a subgroup. An $H$-almost invariant subset $A$ is $H$-proper if it is not equivalent to either the empty set or $G$ modulo finitely many $H$-cosets. Kropholler's conjecture. If $G$ contains an $H$-proper $H$-almost invariant subset $A$ such that $AH=A$, then $G$ admits a nontrivial splitting over a subgroup commensurable with a subgroup of $H$. This generalises Stallings' theorem. The conjecture is known when $H$ and $G$ are Poincaré duality groups, when $H$ is virtually polycyclic, and when $H$ is finitely generated and commensurated in $G$, but remains open in general.