Virtually nilpotent groups' profile equivalence conjecture

Let $G$ be a finitely generated virtually nilpotent group, let $S$ be a finite generating set, and let $\mathcal{D}_{G,S}^{sof}$, $\mathcal{D}_{G,S}^{hyp}$, $\mathcal{D}_{G,S}^{lin}$, and $\mathcal{D}_{G,S}^{fin}$ denote its sofic, hyperlinear, linear sofic, and weakly sofic profiles. Let $\Phi_{G,S}(n)$ denote the full residual finiteness growth. The virtually nilpotent profile conjecture.

$$\mathcal{D}_{G,S}^{sof}(n)\simeq \mathcal{D}_{G,S}^{hyp}(n)\simeq \mathcal{D}_{G,S}^{lin}(n)\simeq \mathcal{D}_{G,S}^{fin}(n)\simeq \Phi_{G,S}(n).$$

This is presented as an ambitious generalization of the corresponding examples for virtually abelian and Heisenberg groups. The weakly sofic part is framed as an open direction, and no resolution is supplied.