Stability conjecture for Steinberg-module homology of congruence subgroups

Let $p$ be a prime. Let $\mathrm{VB}_{\mathbb{Z}}/\Gamma(p)$ denote the indicated module category, let $\mathbfit{A}_{\Gamma(p)}$ be its apartment algebra, let $\Bbbk$ be the coefficient module, and let $\mathbf{St}$ be the Steinberg module. For integers $i,j\geq 0$, consider

$$\operatorname{Tor}^{\mathbfit{A}_{\Gamma(p)}}_i\bigl(\Bbbk,\operatorname{H}_j(\Gamma(p);\mathbf{St})\bigr).$$

Steinberg-module stability conjecture. For each $i,j\geq 0$, the $\mathrm{VB}_{\mathbb{Z}}/\Gamma(p)$-module

$$\operatorname{Tor}^{\mathbfit{A}_{\Gamma(p)}}_i\bigl(\Bbbk,\operatorname{H}_j(\Gamma(p);\mathbf{St})\bigr)$$

is supported in finitely many degrees.

The conjecture is presented as a precise form of the expected representation stability for congruence-subgroup homology with Steinberg coefficients. The paper proves it for $p=3$, $j=0$, and all $i$, while the general assertion remains open.