General geometric conjecture for character-valued parabolic-bundle polynomials

Let ${\bf S}$ be a generic $l$-tuple of regular semisimple adjoint orbits of $\mathfrak{gl}_n(\mathbb{C})$, let $\mathfrak{H}=((\mathfrak{S}_n)^l)\rtimes\mathfrak{S}_l$, and let $\mathcal{X}_{\bf S}^{\mathcal{E}}$ be the associated quotient variety. For $\boldsymbol{\mu}\in(\mathcal{P}_n)^l$, define

$$R_{\boldsymbol{\mu}}=\operatorname{Ind}_{\mathfrak{S}_{\mu^1}}^{\mathfrak{S}_n}(1)\boxtimes\cdots\boxtimes\operatorname{Ind}_{\mathfrak{S}_{\mu^l}}^{\mathfrak{S}_n}(1),$$

and let $\varepsilon^l=\varepsilon\boxtimes\cdots\boxtimes\varepsilon$.

General geometric conjecture. There exists an action of $\mathfrak{H}$ on $H_c^i(\mathcal{X}_{\bf S}^{\mathcal{E}},\mathbb{C})$ such that

$$A_{\boldsymbol{\mu}}^{{\bf b};\mathbf{m}}(t)=t^{-\frac12d_{\bf S}}\sum_i[W_{\boldsymbol{\mu}}^{2i}]t^i,$$

where

$$W_{\boldsymbol{\mu}}^i=\operatorname{Hom}_{(\mathfrak{S}_n)^l}\left(R_{\boldsymbol{\mu}},\varepsilon^l\otimes H_c^{2i}(\mathcal{X}_{\bf S}^{\mathcal{E}},\mathbb{C})\right).$$

The conjecture is the proposed general geometric realization of the character-valued counting polynomial. It is known when $\mathcal{E}=\mathcal{O}(a)^n$, where the source states that the corresponding theorem proves this special case.