Gromov's conjecture on reduced $\ell_p$-cohomology of amenable groups

Let $G$ be an amenable group. For $1<p<\infty$, the reduced $\ell_p$-cohomology of $G$ is the reduced cohomology with coefficients in $\ell_p(G)$, obtained by quotienting the space of cocycles by the closure of the coboundaries.

Gromov's conjecture. The reduced $\ell_p$-cohomology of $G$ vanishes for

$$1<p<\infty.$$

This is a conjecture about vanishing reduced cohomology for amenable groups; the source gives no resolution.