The submonoid conjecture for multiparabolic elements

Let $M$ be a Coxeter matrix over $I$, let $W(M)$ be its Coxeter group with Hecke product $\star$, and let $\tilde{\mathsf P}(M)$ be the set of multiparabolic elements in $W(M)$. Submonoid conjecture. The product set

$$\tilde{\mathsf P}(M)\star\tilde{\mathsf P}(M)$$

is a submonoid of $(W(M),\star)$. The paper places this among conjectures about multiparabolic elements verified in numerous examples; no general resolution is given here.