This question is certainly open in general. I don't know if anyone has formally expressed an 'expectation' in print, but you might be interested in the following pieces of positive evidence.
I believe that all known examples of such 4-manifolds essentially come either from arithmetic constructions or from Coxeter groups. Haglund--Wise showed that all hyperbolic Coxeter groups are cocompactly cubulated (indeed virtually special) , and Bergeron--Haglund--Wise showed that 'standard' cocompact arithmetic lattices in $SO(n,1)$ are virtually special . So most known examples are certainly cocompactly cubulated.
One might worry that closed hyperbolic 3-manifolds are too rigid to be constructed in large numbers, in the way that Kahn--Markovic did with surfaces. But Calegari and I were able to construct very many acylindrical hyperbolic 3-manifolds in random groups . (They don't cubulate for other reasons, but it makes the point that a single group may contain many rigid subgroups.)
 Bergeron, Nicolas(F-PARIS6-IMJ); Haglund, Frédéric(F-PARIS11-M); Wise, Daniel T.(3-MGL)
Hyperplane sections in arithmetic hyperbolic manifolds. (English summary)
J. Lond. Math. Soc. (2) 83 (2011), no. 2, 431–448.
 Calegari, D; Wilton, H.
3-manifolds everywhere, https://arxiv.org/abs/1404.7043
 Haglund, Frédéric(F-PARIS11-M); Wise, Daniel T.(3-MGL)
Coxeter groups are virtually special. (English summary)
Adv. Math. 224 (2010), no. 5, 1890–1903