Galois structure conjecture for torsion cohomology

Let $E/F$ be a cyclic Galois extension of degree $p$, let $E'/F$ be a split extension of degree $p$, and let $\mathcal{M}_\mathcal{U}$ and $\mathcal{M}'_{\mathcal{U}'}$ be the associated locally symmetric spaces with related level structures. Let $\mathcal{L}\rightarrow\mathcal{M}_\mathcal{U}$ and $\mathcal{L}'\rightarrow\mathcal{M}'_{\mathcal{U}'}$ be matching rationally acyclic local systems. The group $\mathbb{Z}[\sigma]$ acts through the Galois permutation, and the cohomology is graded by degree.

Galois structure conjecture. The graded $\mathbb{Z}[\sigma]$-modules

$$H^{*}(\mathcal{M}'_{\mathcal{U}'},\mathcal{L}')\quad\text{and}\quad H^{*}(\mathcal{M}_\mathcal{U},\mathcal{L})$$

are isomorphic.

This conjecture predicts that the rational trace identity for matching level structures extends to torsion cohomology, including the derived-tensor-product behavior in split base-change examples. Its resolution is not given in the source.