The analytic zero-divisor conjecture for torsion-free groups

Let $G$ be a torsion-free group. Write $\alpha \in \mathbb{C}G$ as a finitely supported formal sum and let $f \in \ell^2(G)$; their convolution is

$$\alpha \ast f = \sum_{g,h \in G} a_g b_h gh.$$

Analytic zero-divisor conjecture. If $0 \neq \alpha \in \mathbb{C}G$ and $0 \neq f \in \ell^2(G)$, then

$$\alpha \ast f \neq 0.$$

This is an analytic version of the zero-divisor conjecture and is implied by the strong Atiyah conjecture. Its resolution is not established in the supplied source context.