The analytical zero-divisor conjecture for torsion-free groups

Let $\Gamma$ be a group, let $\mathbf{C}\Gamma$ be its group algebra over $\mathbf{C}$, and let $\lambda_{\Gamma}$ be the left regular representation of $\mathbf{C}\Gamma$ on $\ell^2(\Gamma)$, defined by

$$\lambda_{\Gamma}(f)\xi=f\ast\xi.$$

Analytical zero-divisor conjecture. If $\Gamma$ is torsion-free, then $\lambda_{\Gamma}(f)$ is injective for every non-zero $f\in\mathbf{C}\Gamma$. This is a classical conjecture concerning zero divisors in group algebras and the regular representation; its status is not specified in the source.