Combinatorial identity from a wild-to-tame character-variety correspondence

Let $\mu\in\mathcal{P}_n$, let $\mu'$ be the partition obtained by deleting the first part of $\mu$, and let $n'=n-\mu_1$. Set

$${\bm \mu}'=((n'-1,1),\dots,(n'-1,1),\mu')\in\mathcal{P}_{n'}^{n+1}.$$

Assuming the notation and the wild-to-tame correspondence in the preceding discussion, wild-to-tame combinatorial conjecture.

$$\mathbb{H}_{(\mu,(1^n)),1}(z,w)=\mathbb{H}_{{\bm \mu}'}(z,w).$$

This identity follows in the source from combining Boalch's isomorphism between the relevant wild and tame character varieties with the preceding mixed-Hodge-polynomial conjecture. It is therefore presented as a conjectural combinatorial consequence, rather than an independent theorem.