Heisenberg groups' linear and weakly sofic profile conjecture

Let $H_{2l+1}$ denote the $(2l+1)$-dimensional Heisenberg group, and let $\mathcal{D}_{H_{2l+1},S}^{fin}(n)$ and $\mathcal{D}_{H_{2l+1},S}^{lin}(n)$ denote its weakly sofic and linear sofic profiles relative to a finite generating set $S$. The Heisenberg profile conjecture.

$$\mathcal{D}_{H_{2l+1},S}^{fin}(n)\simeq \mathcal{D}_{H_{2l+1},S}^{lin}(n)\simeq n^{2(2l+1)}.$$

The paper states that the sofic and hyperlinear profiles of these groups already coincide with full residual finiteness growth, and conjectures the analogous conclusion for the weakly sofic and linear sofic profiles.