What is a finite equational basis for this equational theory?

Question

Let our signature be a single binary operation $*$. Let $E$ be the set of all equations $s=t$ such that the set (the set, not the multiset) of variables in the term $s$ is the same as the set of variables in the term $t$. So, for example, the equation $x*y = (y*y)*x$ is in $E$. Let $Th(E)$ be the equational theory generated by $E$. Is there a finite equational basis for that theory? I conjecture that the commutative law, the associative law, and the idempotence law $x*x=x$ are jointly sufficient. Is this true?