Optimal injectivity conjecture for Artin monoid homomorphisms

Let $\mathscr{Ast}$ be the category whose objects are Artin monoids and whose morphisms are compositions of standard homomorphisms. A homomorphism in $\mathscr{Ast}$ is called optimal according to the paper's definition, and is injective on generators when its restriction to the generating set is injective.

Optimal injectivity conjecture. Any optimal and injective on generators homomorphism in $\mathscr{Ast}$ is injective.

This conjecture concerns when compositions of standard homomorphisms preserve injectivity. The paper motivates it by the known injectivity of homogeneous standard homomorphisms and of Tits homomorphisms with all parameters greater than $1$, but does not establish the general assertion.