The injectivity conjecture for relative braid homomorphisms

Let $f:\Sigma\to\Sigma'$ be a morphism as in Lemma, and let $f_*:Br_\Delta^f\to Br_{f(\Delta)}$ be the induced homomorphism. Relative braid homomorphism injectivity conjecture. In the assumptions of Lemma, the homomorphism $f_*$ is injective. In particular, for the canonical homomorphism in Proposition,

$$f_*:Br_\Sigma\to Br_{\Sigma'}$$

is injective. This would extend the injectivity already noted for injective morphisms and establish injectivity for the canonical maps arising from gluing boundary arcs.