Geometric conjecture for character-valued parabolic-bundle polynomials

Let $\mathcal{E}=\mathcal{E}^{{\bf b};\mathbf{m}}$ and let $\boldsymbol{\mu}$ be indivisible. Choose a generic $l$-tuple ${\bf A}$ of adjoint orbits of $\mathfrak{gl}_n(\mathbb{C})$ of type $\boldsymbol{\mu}$, and let $X_{\bf A}^{\mathcal{E}}$ be the associated variety. Let $\mathcal{X}_{\bf A}^{\mathcal{E}}$ denote its categorical quotient by $\operatorname{PAut}(\mathcal{E})$.

Geometric conjecture. (i) The quotient exists as an algebraic variety and $X_{\bf A}^{\mathcal{E}}\to\mathcal{X}_{\bf A}^{\mathcal{E}}$ is a principal $\operatorname{PAut}(\mathcal{E})$-bundle in the étale topology. (ii) The variety $\mathcal{X}_{\bf A}^{\mathcal{E}}$ is nonsingular, has polynomial count, and $H_c^i(\mathcal{X}_{\bf A}^{\mathcal{E}},\mathbb{C})$ has pure mixed Hodge structure. (iii) There is a natural $\mathbb{C}[S_{\boldsymbol{\mu}}]$-module structure on this cohomology such that

$$A_{\boldsymbol{\mu}}^{{\bf b};\mathbf{m}}(t)=t^{-\frac12d_{\bf A}}\sum_i[H_c^{2i}(\mathcal{X}_{\bf A}^{\mathcal{E}},\mathbb{C})]t^i.$$

Here $d_{\bf A}$ is the dimension parameter defined in the source and the bracket denotes the corresponding character-ring element.

This conjecture gives a geometric interpretation of the character-valued polynomial from the preceding conjecture. The source states no resolution for the full three-part assertion.