Trace formula conjecture for logarithmic generating functions and cohomology

Let $w\in S_{\boldsymbol{\mu}}$ and let $D_w=\sum_{i=1}^s\sum_{j=1}^{p_i}\mathfrak{a}_{i,j}$ be compatible with $w$, with $d_{i,j}=\deg(\mathfrak{a}_{i,j})$. For $\boldsymbol{\lambda}\in(\mathcal{P}_n)^r$, choose $\mathbf{v}_{\boldsymbol{\lambda}}=(\mathbf{v}_{i,j})$ with $\mathbf{v}_{i,j}\in(\mathfrak{S}_n)^{d_{i,j}}$ such that $(\overline{\mathbf{v}}_{i,j})$ has cycle type $\boldsymbol{\lambda}$, where $\overline{\mathbf{v}}_{i,j}$ is the product of the coordinates of $\mathbf{v}_{i,j}$.

Trace formula conjecture. One has

$$\left\langle \operatorname{Coeff}_{Y^{\mathbf{m}}}\left[(t-1)\Log\,\Omega_{D_w}(t)\right],p_{\boldsymbol{\lambda}}\right\rangle=t^{-\frac12d_{\bf S}}\sum_i\operatorname{Tr}\left((\mathbf{v}_{\boldsymbol{\lambda}},w)\mid\varepsilon^l\otimes H_c^{2i}(\mathcal{X}_{\bf S}^{\mathcal{E}},\mathbb{C})\right)t^i.$$

This conjecture is presented as evidence and a reformulation related to the preceding general geometric conjecture. The source gives no resolution.