Generalized Tits conjecture for centers of spherical special subgroups

Let $A$ be an Artin group associated to a Coxeter system $(W,S)$. Let $\mathcal S$ be the set of irreducible, spherical subsets of $S$, and let $\operatorname{RA}$ be the right-angled Artin group generated by $\{z_T\}_{T\in\mathcal S}$, with relations

$$[z_T,z_U]=1$$

whenever $U\subseteq T$, $T\subseteq U$, or $[U,T]=1$, where $[U,T]=1$ means that $m_{ut}=2$ for all $u\in U$ and $t\in T$. For each $N$, let $\Phi_N:\operatorname{RA}\to A$ be defined by $\Phi_N(z_T)=\Delta_T^{2N}$, where $\Delta_T^2$ is the center of the corresponding irreducible spherical pure Artin subgroup. Generalized Tits conjecture. The homomorphism $\Phi_N$ is injective for some $N$.

The conjecture extends the Tits conjecture of Crisp and Paris from powers of standard generators to powers of centers of all irreducible spherical special subgroups. It is verified in the paper for locally reducible Artin groups, including all $2$-dimensional Artin groups, and for spherical Artin groups other than types $E_6$, $E_7$, and $E_8$; the full generality remains open.