Zero divisors with small supports in group algebras of torsion-free groups over a field

For any field $\mathbb{F}$ and all torison-free group $\mathbb{G}$, we prove that if $ab=0$ for some non-zero $a,b\in \mathbb{F}[\mathbb{G}]$ such that $\mathrm{\mid }supp(a)\mathrm{\mid }$ $=3$ and $a=1+{\alpha }_1{g}_1+{\alpha }_2{g}_2$, then $g_{1}, g_{2}$satisfiescertainrelationsin$$\mathbb{G}$$.