Monoid parametrization conjecture for units of the noncommutative polygon algebra

Let $\mathbb{T}_n$ be the group generated by the symbols $t_{ij}$ subject to the triangular relations, and let $\mathcal{A}_n$ be the associated algebra with unit group $\mathcal{A}_n^\times$. The assignments $t_{ij}\mapsto x_{ij}$ define a homomorphism $\pi_n:\mathbb{Z}\mathbb{T}_n\twoheadrightarrow\mathcal{A}_n$. Monoid parametrization conjecture. The restriction of $\pi_n$ to $\mathbb{T}_n$ is an isomorphism of monoids

$$\mathbb{T}_n\widetilde\to\mathcal{A}_n^\times.$$

This conjecture would identify all units of $\mathcal{A}_n$ with the distinguished monoid generated by the $x_{ij}$; the source gives no resolution.