The cyclic version of Matsumoto's theorem for infinite parabolic closure

Let $W$ be a Coxeter group and let $C$ be a $W$-conjugacy class. The class satisfies the cyclic version of Matsumoto's theorem (CVMT) if any two reduced words of torically reduced elements in $C$ differ by braid relations and cyclic shifts. For an element $w$, write $\operatorname{supp}(w)$ for its support and $W_{\operatorname{supp}(w)}$ for the corresponding parabolic subgroup; call an irreducible component infinite when the associated Coxeter group is infinite.

Cyclic Matsumoto conjecture. The CVMT holds for a $W$-conjugacy class as long as the parabolic closure $W_{\operatorname{supp}(w)}$ of its minimal elements contains only infinite irreducible components.

The conjecture proposes that failure of CVMT occurs only when a minimal cyclically reduced element has support conjugate to a different support. The supplied context records proofs for broader classes, including infinite-order elements with property $(\operatorname{Cent})$ and finite groups, but gives no resolution of this stated general claim.