Peterson–Thom conjecture for hyperbolic groups

Let $G$ be a hyperbolic group, and let $\mathcal{A}$ and $\mathcal{B}$ be diffuse amenable von Neumann subalgebras of $\mathcal{L}(G)$. Their intersection is diffuse if it contains no minimal projections.

Peterson–Thom conjecture for hyperbolic groups. If $\mathcal{A}\capmathcal{B}$ is diffuse, then the von Neumann algebra they generate is amenable:

$$\mathcal{A}\vee\mathcal{B}\subset \mathcal{L}(G).$$

More precisely, $\mathcal{A}\veemathcal{B}$ is still amenable.

This conjecture is presented as supporting evidence for the paper's amenable-absorption results for groups with hyperbolically embedded subgroups. Unlike the free-group-factor Peterson–Thom conjecture, the supplied text does not state a resolution, so it remains open.