Linnell's analytic zero divisor conjecture for torsion-free groups

Let $G$ be a group, let $\mathbb C G$ be its complex group algebra, and let $\ell^2(G)$ be the Hilbert-space completion of $\mathbb C G$ in the 2-norm. Linnell's analytic zero divisor conjecture. If $G$ is torsion-free, $0\neq\alpha\in\mathbb C G$, and $0\neq\beta\in\ell^2(G)$, then

$$\alpha\beta\neq 0.$$

This analytic formulation strengthens the exclusion of zero divisors by allowing the right factor to lie in $\ell^2(G)$ rather than only in $\mathbb C G$. The source attributes this formulation to Linnell and does not state a general resolution; it is known in connection with special classes of groups but remains open in general.