Brady–McCammond–Mühlherr–Neumann twist-equivalence conjecture for Coxeter generating sets

Let $W$ be a Coxeter group with Coxeter generating sets $S$ and $S'$. Write $S^W$ and $(S')^W$ for their conjugates in $W$, and let $\Gamma(W,S)$ and $\Gamma(W,S')$ denote the corresponding Coxeter graphs. Two Coxeter graphs are twist equivalent when one can be obtained from the other by a sequence of twists.

Brady–McCammond–Mühlherr–Neumann conjecture. If

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

then $\Gamma(W,S)$ is twist equivalent to $\Gamma(W,S')$.

The conjecture asserts that the conjugacy class of the reflections determines the Coxeter graph up to twists. It is presented in the paper as a conjecture of Brady, McCammond, Mühlherr and Neumann; the supplied source gives no resolution evidence.