Generalized Tits injectivity conjecture for Artin monoids

Let $I$ be the index set of a Coxeter matrix, let $M$ be the corresponding Artin monoid, and let $\mathbf d=(d_i)_{i\in I}\in\mathbb Z_{>0}^I$. The associated Tits homomorphism is denoted by $\mathbf T_{\mathbf d}$.

Generalized Tits injectivity conjecture. All Tits homomorphisms $\mathbf T_{\mathbf d}$ are injective.

This extends the Tits conjecture proved by Crisp and Paris in the case where all $d_i>1$, together with related partial results. The assertion is presented as conjectural in the paper.