Regularity conjecture for Burau images of pure braids in $B_4$

Let $B_4$ be the braid group on four strands, let $\rho$ denote the representation used in the paper, and let $T_4$ be the object on which the braid acts. For a Laurent polynomial, write its lowest degree for the least exponent of its variable. Let $\Delta$ and $a$ be the braids appearing in the statement, and let $\sigma\in B_4$ be a non-trivial pure braid not equivalent to $\Delta^m$ for any $m\in\mathbb{N}$. Assume that $\sigma$ acts non-trivially on $T_4$. Regularity conjecture. There exist sufficiently large $l_0\in\mathbb{N}$, depending on the length of $\sigma$, and sufficiently large $m_0\in\mathbb{N}$ such that, for every $m>m_0$ and $l\ge l_0$, the Laurent polynomials $\rho_{11}(a^m\sigma a^{-l})$ and $\rho_{31}(a^m\sigma a^{-l})$ are non-zero and the difference of their lowest degrees is equal to $-1$. The conjecture records an experimentally observed regularity in these matrix entries of Burau images; its resolution would clarify the structure of images of braids of this special form, but no resolution is supplied here.