Hartley's conjecture on group identities in units of group algebras

Let $G$ be a torsion group, let $F$ be a field, and let $FG$ be the group algebra of $G$ over $F$. A group $H$ satisfies a group identity if there is a nontrivial group word $w$ such that $w(h_1,\ldots,h_n)=1$ for all $h_1,\ldots,h_n\in H$. An algebra satisfies a polynomial identity if there is a nonzero polynomial in noncommuting variables that vanishes under every substitution from the algebra.

Hartley's conjecture. If the unit group $U(FG)$ satisfies a group identity, then $FG$ satisfies a polynomial identity.

This conjecture proposes a general relationship between group identities in unit groups and polynomial identities in group algebras. The supplied context attributes it to Brian Hartley but gives no evidence of whether it has been resolved.