Peterson–Thom conjecture on unique maximal amenable extensions

Let $r>1$ and let $\mathcal{Q}$ be a diffuse amenable von Neumann subalgebra of the free group factor $\mathcal{L}(\mathbb{F}_r)$. A von Neumann subalgebra $\mathcal{P}$ is maximal amenable if it is amenable and is not properly contained in any larger amenable von Neumann subalgebra of $\mathcal{L}(\mathbb{F}_r)$.

Peterson–Thom conjecture. There exists a unique maximal amenable von Neumann subalgebra $\mathcal{P}$ of $\mathcal{L}(\mathbb{F}_r)$ such that

$$\mathcal{Q}\subset \mathcal{P}.$$

A positive answer would imply major structural properties of free group factors, including primeness, the absence of Cartan subalgebras, and strong solidity. The conjecture was proved by Belinschi and Capitaine, independently by Bordevane and Collins, and by other authors, so its database status is solved.