I am trying to prove the following:
"A non-positively curved cube complex $X$ that admits a local isometric embedding (that maps cubes to cubes) into the Salvetti complex of some right-angled Artin group is special."
By "special" we mean that hyperplanes are embedded and 2-sided and that they do not self-osculate or inter-osculate (hopefully, this is the usual definition).
I think that one might show that the Salvetti complex itself is special and then that the condition of $X$ being locally isometrically embedded pulls back the "specialness" property on $X$, but I cannot see how this last part should be done, and I would be glad to receive any hint.
The key point to notice is that a combinatorial map $f: X \to Y$ sends two edges dual to the same hyperplane either to a single edge or to two edges dual to the same hyperplane. Thus, if $f$ is moreover locally injective and if $e_1,e_2$ are two edges with a common endpoint and dual to the same hyperplane, then $f(e_1)$ and $f(e_2)$ are two distinct edges of $Y$ dual to the same hyperplane. As a consequence, $f$ sends self-intersecting hyperplanes to self-intersecting hyperplanes. The same argument essentially holds for the other pathological configurations of hyperplanes.