Equality of upper and lower sofic lp-dimensions

Let $2<p<\infty$, let $\Gamma$ be a countable discrete sofic group, and let $\Sigma$ be a sofic approximation of $\Gamma$. Equality conjecture. For every $n\in\mathbb N$,

$$\dim_{\Sigma,l^{p}}(l^{p}(\Gamma)^{\oplus n},\Gamma)=\underline{\dim}_{\Sigma,l^{p}}(l^{p}(\Gamma)^{\oplus n},\Gamma).$$

This conjecture concerns whether the upper and lower sofic $l^{p}$-dimensions coincide in the range $2<p<\infty$. The paper says that little progress has been made on it and suggests that the definition might not be the right analogue of von Neumann dimension in this range.