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.