  In this paper, we build on the work from our previous paper
(arXiv:2002.01556) to show that periodic rational $G$-equivariant topological
$K$-theory has a unique genuine-commutative ring structure for $G$ a finite
abelian group. This means that every genuine-commutative ring spectrum whose
homotopy groups are those of $KU_{\mathbb{Q},G}$ is weakly equivalent, as a
genuine-commutative ring spectrum, to $KU_{\mathbb{Q},G}$. In contrast, the
connective rational equivariant $K$-theory spectrum does not have this type of
uniqueness of genuine-commutative ring structure.
