Mühlherr's sharp-angled automorphism conjecture for Coxeter generating sets

Let $W$ be a Coxeter group with Coxeter generating sets $S$ and $S'$ such that

$$S^W=(S')^W.$$

A reflection-preserving automorphism of $W$ is an automorphism that preserves the set of reflections. A set is sharp-angled with respect to $S'$ when it has the sharp-angled property relative to that Coxeter generating set.

Mühlherr's conjecture. There exists a reflection-preserving automorphism $\alpha$ of $W$ such that $\alpha(S)$ is sharp-angled with respect to $S'$.

This conjecture seeks a controlled relation between Coxeter generating sets having the same conjugacy-invariant reflection data. The supplied source gives no evidence that it has been resolved.