A note on semicompleteness of graph products of abelian groups

In this short note we prove that a graph product $G_\Gamma$ of finitely generated abelian groups is semicomplete -- that is the kernel of the natural homomorphism ${\rm Aut}(G_\Gamma)\to{\rm Aut}(G_\Gamma^{ab})$ induced by the abelianization of $G_\Gamma$ is equal to the inner automorphisms -- if and only if $\Gamma$ does not have a separating star.