Vaes–Wahl conjecture on weakly mixing Maharam extensions of Bernoulli actions

Let $G$ be a countable group with non-vanishing first $\ell^2$-cohomology. Equivalently, suppose that there is a function $f:G\to\mathbb{C}$ with $f(0)=0$, which is not in $\ell^2(G)$, but satisfies

$$\sum_{h\in G}\bigl(f(gh)-f(h)\bigr)^2<\infty,\qquad\text{for all }g\in G.$$

A non-singular Bernoulli action is an action $G\curvearrowright(\{0,1\}^G,\mu)$ preserving the measure class of the product probability measure. Vaes–Wahl's conjecture. There is such a non-singular Bernoulli action whose Maharam extension is ergodic, or even weakly mixing.

This conjecture concerns which type $\mathrm{III}_\lambda$ behaviours can occur for non-singular Bernoulli actions of general countable groups. The cited work of Vaes and Wahl is described as making substantial progress on this assertion, but the supplied text does not establish its full resolution.