The $S^{p}$-dimension conjecture for noncommutative $L^{p}$ spaces

Let $\Gamma$ be an $\mathcal{R}^{\omega}$-embeddable group, fix $2<p<\infty$, and consider the multiplication action of $\Gamma$ on $L^{p}(L(\Gamma),\tau_{\Gamma})^{\oplus n}$. Let $\Sigma$ be an embedding sequence. The noncommutative $L^{p}$-dimension conjecture.

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

The author notes that the preceding method does not compute $S^{p}$ dimension for $1\leq p<2$ and that proving this claim requires a sufficiently strong lower bound for approximate dimensions.