Interlacing conjecture for deformed monotone Hurwitz numbers

Let $g\geqslant 0$, $n\geqslant 1$, and let $\mu_1,\ldots,\mu_n\geqslant 1$. The deformed monotone Hurwitz number $\vec{H}^t_{g,n}(\mu_1,\ldots,\mu_n)$ is a polynomial in $t$. Two real-rooted polynomials interlace when their roots weakly alternate, with their degrees differing by one. Interlacing conjecture. The polynomial $\vec{H}^t_{g,n}(\mu_1,\ldots,\mu_n)$ interlaces each of the $n$ polynomials

$$\vec{H}^t_{g,n}(\mu_1+1,\mu_2,\ldots,\mu_n),\quad \vec{H}^t_{g,n}(\mu_1,\mu_2+1,\ldots,\mu_n),\quad\ldots,\quad \vec{H}^t_{g,n}(\mu_1,\mu_2,\ldots,\mu_n+1).$$

For fixed $(g,n)$ this predicts a lattice of interlacing polynomials; it has been checked computationally in many cases, but remains unproved in general.