Putman's non-isomorphism conjecture for twisted cohomology maps

Let $r\ge 2$, and consider the map

$$H^{k-r}(\operatorname{\textup{Mod}}_{g,p}^b;H^1(\Sigma_{g,p}^b;\operatorname{\mathbb{Q}})^{\otimes r}) \longrightarrow H^{k-r}(\operatorname{\textup{Mod}}_{g,p}^b(\ell);\operatorname{\mathfrak{H}}_{g,p}^b(\ell;\operatorname{\mathbb{Q}})^{\otimes r}).$$

Here the map is induced by the covering map from the regular cover $S_{\mathcal{D}}\to\Sigma_{g,p}^b$, where $\mathcal{D}=H_1(\Sigma_g;\mathbb{Z}/\ell)$ and $\operatorname{\mathfrak{H}}_{g,p}^b(\ell;\operatorname{\mathbb{Q}})=H^1(S_{\mathcal{D}};\operatorname{\mathbb{Q}})$. Putman's conjecture. For $r\ge 2$, the map is not an isomorphism. Putman's result proves that the analogous map is an isomorphism for $r=1$ in a stable range, and the conjecture predicts that this phenomenon does not extend to tensor powers of degree at least two.