Gournay dimension-zero characterization by invariant exhaustions

Let $G$ be a countable discrete amenable group, let $1\leq p<\infty$, and let $F\subseteq \ell^pG$ be a closed invariant subspace. An invariant exhaustion $(E_i)_i$ of $\ell^pG$ is an exhaustion by invariant subspaces, and $\dim_{\ell^p}F$ denotes the $\ell^p$-dimension of $F$.

Gournay's dimension-zero conjecture. One has

$$\dim_{\ell^p}F=0$$

if and only if there is an invariant exhaustion $(E_i)_i$ of $\ell^pG$ such that

$$E_i\cap F=0$$

for all $i$.

The paper notes that the existence of such an exhaustion implies $\dim_{\ell^p}F=0$ for amenable groups and conjectures the converse. The analogous question can also be considered for sofic groups; the source does not provide a resolution.