Let KG be the group algebra of a torsion group G over a field K. We show that if the units of KG satisfy a Laurent polynomial identity, which is not satisfied by the units of the relative free algebra K[α,β:α2=β2=0] , then KG satisfies a polynomial identity. This extends Hartley’s Conjecture which states that if the units of KG satisfy a group identity, then KG satisfies a polynomial identity. As an application we prove that if the units of KG satisfy a Laurent polynomial identity whose support has cardinality at most 3, then KG satisfies a polynomial identity.