Ordered-triangular extension conjecture for irreducible three-strand braid-group representations

Let $\rho$ be an irreducible $d$-dimensional matrix representation of the braid group $\mathcal B_3$, with $\rho(\sigma_1)=A$ and $\rho(\sigma_2)=B$. The matrices are in ordered triangular form when $A$ is upper triangular, $B$ is lower triangular, and

$$B_{i,i}=A_{d-i+1,d-i+1}.$$

A standard extension is an extension of $\rho$ to the loop braid group $\mathcal{LB}_3$ for which the image of the relevant generator satisfies $S=kAB$ for some $k\in\mathbb C$. Ordered-triangular extension conjecture. If $\rho$ is irreducible and $A$ and $B$ are in ordered triangular form, then $\rho$ has a standard extension to $\mathcal{LB}_3$. The conjecture is motivated by the explicit construction of ordered-triangular representations and evidence from low-dimensional cases, but the general assertion remains open.