Vanishing of finite-dimensional representations in infinite sofic groups

Let $\Gamma$ be an infinite sofic group, let $\Sigma$ be a sofic approximation, and let $V$ be a finite-dimensional representation of $\Gamma$. The finite-dimensional vanishing conjecture. For every $1\leq p<\infty$,

$$\dim_{\Sigma,l^{p}}(V,\Gamma)=0.$$

The paper says this is suggested by its analysis of trivial actions; it also remarks that an analogous statement should hold for $\mathcal{R}^{\omega}$-embeddable groups with $l^{p}$ replaced by $S^{p}$.