Gornay's dimension conjecture for amenable groups

Let $\Gamma$ be an amenable group and let $Y\subseteq l^{p}(\Gamma)^{\oplus n}$ for some $n\in \mathbb N$. Let $\dim_{l^{p}}^{G}(Y,\Gamma)$ denote the $l^{p}$-dimension defined by Gornay, and let $\Sigma$ be a sofic approximation of $\Gamma$. Gornay's dimension conjecture. For every sofic approximation $\Sigma$ of $\Gamma$,

$$\dim_{l^{p}}^{G}(Y,\Gamma)=\dim_{\Sigma,l^{p}}(Y,\Gamma).$$

This conjecture asks whether Gornay's dimension agrees with the sofic $l^{p}$-dimension for amenable groups. The paper states that little progress has been made on it.