Letellier's conjecture on character-variety polynomials

For a partition $\lambda$ of $n$, let $C_\lambda$ be the unipotent conjugacy class of $\operatorname{GL}_n(\mathbb{C})$ whose Jordan-block sizes are given by the dual partition $\lambda'$. For a multi-partition $\bm{\mu}=(\mu^1,\dots,\mu^k)$ of $n$, define

$$\overline{C}_{\bm{\mu}}:=\operatorname{GL}_n^{2g}\times \overline{C}_{\mu^1}\times\cdots\times \overline{C}_{\mu^k}.$$

Fix a primitive $n$-th root of unity $\zeta$, and let ${\mathcal{Z}}_{\bm{\mu}}$ be the space of tuples

$$(A_1,\dots,A_g,B_1,\dots,B_g,X_1,\dots,X_k)\in\overline{C}_{\bm{\mu}}$$

satisfying

$$\prod_{i=1}^g(A_i,B_i)\prod_{j=1}^kX_j=\zeta\cdot I_n,$$

where $(A,B)=ABA^{-1}B^{-1}$. Put

$${\mathcal{M}}_{\bm{\mu}}:={\mathcal{Z}}_{\bm{\mu}}/\!/\operatorname{GL}_n=\operatorname{Spec}\left(\mathbb{C}[{\mathcal{Z}}_{\bm{\mu}}]^{\operatorname{GL}_n}\right).$$

Let $PP_c({\mathcal{M}}_{\bm{\mu}},t):=\sum_s ih_c^{s,s;2s}({\mathcal{M}}_{\bm{\mu}})t^s$ be the pure part of the compactly supported mixed Poincaré polynomial, and let $V_{\bm{\mu}}(t)$ and $d_{\bm{\mu}}$ be as above. Letellier's conjecture.

$$V_{\bm{\mu}}(t)=t^{-d_{\bm{\mu}}/2}PP_c({\mathcal{M}}_{\bm{\mu}},t).$$

This conjecture proposes the interpretation of the polynomials $V_{\bm{\mu}}(t)$ as pure compactly supported intersection-cohomology polynomials of character varieties. It is attributed in the source to Letellier, and the status is not determined by the provided text.