Sofic and hyperlinear profile comparison

Let $G$ be a sofic group. The sofic profile $athcal{D}_{G}^{sof}(n)$ and hyperlinear profile $athcal{D}_{G}^{hyp}(n)$ are compared using the relation $reccurlyeq$. The profile comparison conjecture.

$$\mathcal{D}_{G}^{sof}(n)\preccurlyeq \mathcal{D}_{G}^{hyp}(n).$$

This would give the reverse asymptotic comparison to the known bound from sofic approximations into unitary groups. Its status is not resolved in the supplied text.