Virtual Specialness of certain graphs of special cube complexes

We investigate the virtual specialness of a compact cube complex $X$ that splits as a graph of nonpositively curved cube complexes. We prove virtual specialness of $X$ when each vertex space of $X$ has word-hyperbolic $\pi_1$ and $\pi_1X$ has ``finite stature'' relative to its edge groups. The results generalize the motivating case when tree $\times$ tree lattices are virtual products.