The relative solidity conjecture for relatively hyperbolic groups

Let $G$ be a group that is hyperbolic relative to a finite family of subgroups $\{H_1,\ldots,H_n\}$. Denote by $\mathcal M=\mathcal L(G)$ the von Neumann algebra corresponding to $G$. Let $p\in\mathcal M$ be a nonzero projection and let $\mathcal A\subseteq p\mathcal M p$ be a von Neumann subalgebra whose relative commutant $\mathcal A'\cap p\mathcal M p$ has no amenable direct summand. Relative solidity conjecture. There is some $1\leq i\leq n$ such that a corner of $\mathcal A$ intertwines into $\mathcal L(H_i)$ inside $\mathcal M$, in the sense of Popa; that is, $\mathcal A\prec_{\mathcal M}\mathcal L(H_i)$. This concerns the structure of commuting subalgebras of group von Neumann algebras associated with relatively hyperbolic groups; the supplied text gives no resolution status for the conjecture.