Let $K$ be a field and $G$ be a group. Then $K[G]$ is a domain iff $G$ is torsion-free. I know that "$\Leftarrow$" is conjectured to be always true. But what about the other direction?
Suppose $G$ is not torsion-free and $g\in G$ has order $n$ for some integer $n>1$. We know that $g^n=e$ and:
so $K[G]$ is not a domain. (I guess the whole point of the conjecture is whether or not this is the only general example of zero-divisors in the group ring.)
I hope this helps!