Integer-solution conjecture for commensurable 2-dimensional right-angled Artin groups

Let $\mathbb{G}(\Gamma_1)$ and $\mathbb{G}(\Gamma_2)$ be $2$-dimensional right-angled Artin groups, meaning that their commutation graphs are triangle-free. Let $S_i(\Gamma_1,\Gamma_2)$ be the system defined by the product graph. Integer-solution conjecture. If $\mathbb{G}(\Gamma_1)$ and $\mathbb{G}(\Gamma_2)$ are commensurable, then $S_i(\Gamma_1,\Gamma_2)$ has integer solutions. This is presented as a proposed generalisation of the reduction from commensurability to integer solutions for RAAGs defined by trees. Establishing it would provide a starting point for extending the authors' gluing strategy to $2$-dimensional RAAGs; the supplied text gives no evidence that it has been resolved.