Virtually abelian groups' linear profile conjecture

Let $G$ be a finitely generated virtually abelian group, and let $\operatorname{rank} G$ denote its rank. Write $\mathcal{D}_{G,S}^{lin}(n)$ for the linear sofic profile of $G$ relative to a finite generating set $S$, and use $\simeq$ for the profile-equivalence relation. The virtually abelian linear-profile conjecture.

$$\mathcal{D}_{G,S}^{lin}(n)\simeq n^{\operatorname{rank} G}.$$

The source explains that this is motivated by known stability results, but that stability of virtually abelian groups with respect to the rank metric remains unknown; a partial result is cited there.