Sofic-approximation independence of lp-dimension for amenable groups

Let $\Gamma$ be an amenable group, let $\Sigma$ and $\Sigma'$ be two sofic approximations of $\Gamma$, and let $X$ be a uniformly bounded representation of $\Gamma$. Approximation-independence conjecture.

$$\dim_{\Sigma,l^{p}}(X,\Gamma)=\dim_{\Sigma',l^{p}}(X,\Gamma).$$

This conjecture asks whether the sofic $l^{p}$-dimension of a uniformly bounded representation is independent of the chosen sofic approximation for amenable groups. The source presents it as a natural conjecture based on the techniques developed in the paper.