Classification of right-angled Coxeter groups with a strongly solid von Neumann algebra

Let $W$ be a finitely generated right-angled Coxeter group with group von Neumann algebra $\mathcal{L}(W)$. We prove the following dichotomy: either $\mathcal{L}(W)$ is strongly solid or $W$ contains $\mathbb{Z}\times {\mathbb{F}}_2$ as a subgroup. This proves in particular strong solidity of $\mathcal{L}(W)$ for all non-hyperbolic Coxeter groups that do not contain $\mathbb{Z}\times {\mathbb{F}}_2$.