Lewis's Hurwitz-orbit classification conjecture for finite Coxeter groups

Let $W$ be a finite Coxeter group and let $g\in W$ be an arbitrary element. A reflection factorization of $g$ is a tuple of reflections whose product is $g$, and the Hurwitz action acts on such factorizations by the standard Hurwitz moves. For a factorization $(g_1,\ldots,g_k)$, let

$$H=\langle g_1,\ldots,g_k\rangle\leq W$$

and consider the multiset of $H$-conjugacy classes of its factors.

Lewis's conjecture. Two reflection factorizations of $g$ belong to the same Hurwitz orbit if and only if they generate the same subgroup $H\leq W$ and have the same multiset of $H$-conjugacy classes.

The product, generated subgroup, and multiset of conjugacy classes are evident invariants of the Hurwitz action. The conjecture was originally posed for complex reflection groups, but the source notes counterexamples for minimum-length reflection factorizations in two exceptional complex reflection groups; the finite Coxeter-group formulation is the claim considered here.