Injectivity conjecture for minimal infinite reduced words

Let $\mathbf i$ be an infinite reduced word, and let $e_{\mathbf i}$ be its associated factorization map. Say that $\mathbf i$ is minimal in its block when it is minimal in the relevant braid-equivalence block. Injectivity conjecture. If $\mathbf i$ is an infinite reduced word which is minimal in its block, then $e_{\mathbf i}$ is injective. The paper proves non-injectivity for words that are not minimal in their block, while the minimal case is presented as an open conjecture.