Embedding criterion for right-angled Artin groups in Brin-Thompson groups

Let $\Gamma$ be a finite simplicial graph, and let $A_\Gamma$ be the right-angled Artin group defined by generators corresponding to the vertices of $\Gamma$, with two generators commuting exactly when the corresponding vertices are adjacent. Let $n\geq 1$, and write $nV$ for the $n$-dimensional Brin-Thompson group. Embedding criterion. The group $A_\Gamma$ embeds into $nV$ if and only if it does not contain $\mathbb{Z}^{n+1}*\mathbb{Z}$. This would generalize the known case for $n=1$, where the only right-angled Artin groups embedding into Thompson's group $V$ are direct products of free groups; the criterion is presented as open in the source.