Genus-asymptotic roots conjecture for deformed monotone Hurwitz numbers

Fix positive integers $\mu_1,\ldots,\mu_n$ whose sum is $d$. Let the roots of the polynomial $\vec{H}^t_{g,n}(\mu_1,\ldots,\mu_n)$ be ordered from smallest to largest. Genus-asymptotic roots conjecture. As $g\to\infty$, these roots respectively approach

$$-\frac{d-2}{1},\quad-\frac{d-3}{2},\quad-\frac{d-4}{3},\quad\ldots,\quad-\frac{2}{d-3},\quad-\frac{1}{d-2},\quad 0.$$

Moreover, convergence to a number less than $-1$ is increasing from below, while convergence to a nonzero number greater than $-1$ is decreasing from above. This conjecture is supported by extensive data, but remains open.