The left-ideal lower-bound conjecture for $S^{p}$ dimension

Let $\Gamma$ be an $\mathcal{R}^{\omega}$-embeddable group, let $2\leq p<\infty$, and let $C^{*}_{\lambda}(\Gamma)$ be the reduced $C^{*}$-algebra. Let $I\subseteq C^{*}_{\lambda}(\Gamma)$ be a norm-closed left ideal. Regard $C^{*}_{\lambda}(\Gamma)$ as a subalgebra of $L(\Gamma)$, and let $q\in L(\Gamma)$ be the projection satisfying

$$\overline{I}^{\operatorname{wk}^{*}}=L(\Gamma)q.$$

The left-ideal lower-bound conjecture. For the action of $\Gamma$ by left multiplication,

$$\dim_{\Sigma,S^{p},\operatorname{multi}}(I,\Gamma)\geq \tau(q).$$

This is proposed as a noncommutative analogue of the preceding Fourier-based lower bound; the paper gives no proof and leaves it as a conjecture.