Thin invariant exhaustion conjecture for infinite groups

Let $G$ be an infinite group. An invariant exhaustion of $\ell^pG$ is an increasing sequence of invariant subspaces whose union is dense in $\ell^pG$, and it is thin if there is a closed invariant subspace $F\neq 0$ such that $E_i\cap F=0$ for every member $E_i$ of the exhaustion.

Thin invariant exhaustion conjecture. Every infinite group $G$ admits a thin invariant exhaustion of $\ell^p G$ for all $p>2$.

The theorem preceding this conjecture proves the assertion for infinite elementary amenable groups, while the paper extends it to groups containing an infinite elementary amenable subgroup. The conjecture asks whether the conclusion holds for every infinite group.