We study the simplicial volume of manifolds obtained from Davis' reflection group trick, the goal being characterizing those having positive simplicial volume. In particular, we focus on checking whether manifolds in this class with nonzero Euler characteristic have positive simplicial volume (Gromov asked whether this holds in general for aspherical manifolds). This leads to a combinatorial problem about triangulations of spheres: we define a partial order on the set of triangulations -- the relation being the existence of a nonzero-degree simplicial map between two triangulations -- and the problem is to find the minimal elements of a specific subposet. We solve explicitly the case of triangulations of the two-dimensional sphere, and then perform an extensive analysis, with the help of computer searches, of the three-dimensional case. Moreover, we present a connection of this problem with the theory of graph minors.