The semicompleteness characterization for graph products of abelian groups

Let $\Gamma$ be a finite graph with vertex groups $G_1,\ldots,G_n$, where each $G_i$ is an abelian group, and let $G_\Gamma$ be their graph product. The graph $\Gamma$ has a separating star if the complement of the star of some vertex is disconnected.

Semicompleteness characterization. The graph product $G_\Gamma$ is semicomplete if and only if $\Gamma$ has no separating star.

The statement extends the result established in the paper for finitely generated abelian vertex groups to arbitrary abelian vertex groups. Its status is not resolved in the supplied text.