The Diophantine problem conjecture for star-shaped right-angled Baumslag–Solitar–Artin groups

Let $G$ be a group, and let $B(\Gamma)$ be a right-angled Baumslag–Solitar–Artin group whose underlying undirected graph is a star graph. Suppose that $G$ is virtually $B(\Gamma)$, meaning that $G$ has a finite-index subgroup isomorphic to $B(\Gamma)$. Diophantine decidability conjecture. The Diophantine problem is decidable in $G$. This extends the known result for groups virtually direct products of hyperbolic groups, and in particular covers star-shaped right-angled Baumslag–Solitar–Artin groups; the conjecture remains open in general.