Multiplicity stability conjecture for $VIC(\mathbb{Z}/p^k\mathbb{Z})$-modules

For a prime $p$ and a positive integer $k$, consider the irreducible complex representations of $GL_n(\mathbb{Z}/p^k\mathbb{Z})$ for all $n$, together with a compatible collection of labelings. A finitely generated $VIC(\mathbb{Z}/p^k\mathbb{Z})$-module is a functor with finite generation under the morphisms of $VIC(\mathbb{Z}/p^k\mathbb{Z})$. Multiplicity stability conjecture. For every $p,k$ there exists a compatible collection of labelings of the irreducible complex representations of $GL_n(\mathbb{Z}/p^k\mathbb{Z})$ for all $n$ such that finitely generated $VIC(\mathbb{Z}/p^k\mathbb{Z})$-modules exhibit multiplicity stability with respect to this labeling. This would extend the known multiplicity-stability phenomenon for $VIC(\mathbb{F}_p)$-modules, but the source notes that suitable labelings for the higher prime-power groups are not known.