Fourier lower-bound conjecture for sofic $l^{p}$ dimension

Let $\Gamma$ be a sofic group and fix $1\leq p\leq 2$. Set $M=L(\Gamma)$, let $\tau$ be the canonical group trace, and let $p'$ satisfy $1/p+1/p'=1$. Let $\mathcal{F}\colon l^{p}(\Gamma)\to L^{p'}(M,\tau)$ be the Fourier transform. For $X\subseteq l^{p}(\Gamma)$, let $q\in M$ be a projection satisfying

$$\overline{\mathcal{F}(X)}^{\|\cdot\|_{p}}=L^{p}(M,\tau)q.$$

The Fourier lower-bound conjecture.

$$\underline{\dim}_{\Sigma,l^{p}}(X,\Gamma)\geq \tau(g).$$

Moreover, if $\Gamma$ is $\mathcal{R}^{\omega}$-embeddable, with the same $X,p,p',q,\tau$, then

$$\underline{\dim}_{\Sigma,S^{p},\operatorname{conj}}(X,\Gamma)\geq \tau(q).$$

The source itself warns that the conjecture may be too strong in full generality, while suggesting that special cases could still be meaningful; the first displayed bound contains $\tau(g)$ although the setup defines $q$.