Kourovka Notebook Problem 21.128

Two groups $G_1$ and $G_2$ are said to be commensurable if there exist finite-index subgroups $H_1\leqslant G_1$ and $H_2\leqslant G_2$ (not necessarily of the same index) such that $H_1\cong H_2$. Let $A[F_4]$ and $A[H_4]$ denote the Artin groups of spherical types $F_4$ and $H_4$, respectively. Are these two groups commensurable?