Virtual specialness for graphs of virtually special cube complexes with cyclonormal edges

Let $X$ be a compact cube complex that splits as a graph of virtually special cube complexes with cyclonormal edges. Virtual-specialness conjecture. Then $X$ is virtually special. This would extend the established combination result beyond hyperbolic vertex groups. The paper proves the associated finite-stature statement under these hypotheses, while this converse virtual-specialness assertion is presented as expected and remains open.