  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 $K[\alpha,\beta :
\alpha^2=\beta^2=0]$ then $KG$ satisfies a polynomial identity. This extends
Hartley Conjecture which states that if the units of $KG$ satisfies a group
identity then $KG$ satisfies a polynomial identity. As an application of our
results we prove that if the units of $KG$ satisfies a Laurent polynomial
identity with a support of cardinality at most 3 then $KG$ satisfies a
polynomial identity.
