Let $K$ be a finite field and $G$ be a discrete group.

Is it true that for every $a=e+a_1+\ldots+a_n,b=e+b_1+\ldots+b_m\in K[G]$ with $b_i\neq e,a_j\neq e$ the condition $ab=0$ implies $ba=0$?

It is related to this question. The examples proposed there do not violate the condition above.