Periodic stability for homology of congruence subgroups

Let $\Gamma_n(p)$ and $\Gamma^{\operatorname{Sp}}_{2n}(p)$ denote the congruence subgroups considered in the source, with homology over $\mathbb{F}_p$ carrying the natural actions of $\operatorname{SL}_n(\mathbb{F}_p)$ and $\operatorname{Sp}_{2n}(\mathbb{F}_p)$. Modular periodic stability conjecture. For every $i\geq 0$ and every prime $p$, the sequence $\{H_i(\Gamma_n(p);\mathbb{F}_p)\}$ is uniformly mixed tensor stably periodic with period $p$ as a sequence of $\operatorname{SL}_n(\mathbb{F}_p)$-representations, and $\{H_i(\Gamma^{\operatorname{Sp}}_{2n}(p);\mathbb{F}_p)\}$ is uniformly stably periodic with period $p$ as a sequence of $\operatorname{Sp}_{2n}(\mathbb{F}_p)$-representations. This conjecture proposes periodic analogues of representation stability for congruence-subgroup homology; the source gives no resolution.