Minimal vertex counts for constant-curvature polyhedra

A constant-curvature polyhedron (CCP) is a polyhedron whose faces have constant Gaussian curvature. Consider CCPs realizing closed surfaces, classified by orientability and topological genus $g$, and let the vertex count mean the number of vertices of the polyhedron. For orientable CCPs, self-intersections may either be excluded or permitted as specified.

Minimal vertex-count conjecture. (i) For a CCP of a topological torus without self-intersection, the minimal number of vertices is $9$. (ii) For an orientable CCP of genus $g$, if self-intersections are permitted, then the minimal number of vertices is $2g+4$.

Known examples suggest that the listed constructions for orientable CCPs without self-intersections and for non-orientable CCPs are not necessarily minimal in vertex count. The conjecture asserts the stated minima for the torus and for orientable surfaces when self-intersections are allowed.