Group algebras with units satisfying a group identity

Let $K[G]$ be the group algebra of a group $G$ over a field $K$, and let $U(K[G])$ be its group of units. A conjecture by Brian Hartley asserts that if $G$ is a torsion group and $U(K[G])$ satisfies a group identity, then $K[G]$ satisfies a polynomial identity. This was verified earlier in case $K$ is an infinite field. Here we modify the original proof so that it handles fields of all sizes.