Direct finiteness conjecture for group algebras of countable groups

Let $F$ be a field and $G$ a countable group. The group algebra $F(G)$ is directly finite, meaning that for any $x,y\in F(G)$, $xy=1$ implies $yx=1$. Direct finiteness conjecture. For any field $F$ and any countable group $G$, the group algebra $F(G)$ is directly finite. This is a stable-finiteness question for group algebras; the supplied source span does not indicate whether the assertion is proved or remains open.