Rigidity conjecture for special mixed graphs without satellites

Let $\Gamma$ be a special mixed graph. A satellite of a sinkhole $a$ is a vertex at distance $2$ from $a$ such that every vertex joined to it is also joined to $a$. The associated T-RAAG $T(\Gamma)$ is called rigid when its isomorphism type determines the associated mixed graph.

Rigidity conjecture. If $\Gamma$ has no satellites, then it is rigid.

The paper proves that rigidity implies the absence of satellites for special mixed graphs, and proves the converse for Droms mixed graphs and for special mixed graphs having a sinkhole joined to every other vertex. The conjecture asks whether the converse holds for all special mixed graphs.