Kropholler's conjecture on almost invariant sets and tree actions

Let $G$ be a group and let $H$ be a subgroup. A subset $S\subset G$ is $H$-finite if it is contained in finitely many right cosets of $H$. Two subsets are $H$-almost equal if their symmetric difference is $H$-finite, and a set $A$ is $H$-almost invariant if $A$ is $H$-almost equal to $Ag$ for every $g\in G$. Such a set is proper if neither $A$ nor $G\setminus A$ is $H$-finite.

Kropholler's conjecture. If there is a proper $H$-almost invariant set $A$ such that $A=AH$, then $G$ has a non-trivial action on a tree in which $H$ fixes a vertex $v$ and every edge incident with $v$ has an $H$-finite stabiliser.

This is a relative form of Stallings' Ends Theorem concerning the action of a group on a tree. The source presents the statement as the Kropholler Conjecture and describes the paper as attempting a complete proof, but the supplied status is unknown; the resolution should therefore be checked.