The maximal FAn subgroup conjecture for Coxeter groups

Let $(W,S)$ be a Coxeter system. A subgroup $H\subset W$ is maximal $\operatorname{FA}_n$ if it has property $\operatorname{FA}_n$ and is not properly contained in another subgroup of $W$ with property $\operatorname{FA}_n$. A special subgroup is $(n+1)$-spherical if every special subgroup of it of rank at most $n+1$ is finite.

Maximal FAn subgroup conjecture. A subgroup $H\subset W$ is maximal $\operatorname{FA}_n$ if and only if

$$H=wAw^{-1}$$

for some maximal $(n+1)$-spherical special subgroup $A$ of $W$ and some $w\in W$.

The conjecture is known for $n=1$, by work of Mihalik and Tschantz. Its status in higher dimensions is not resolved in the supplied source.