The geometric realization conjecture for triangulated closed orientable surfaces of genus at most four

A triangulation is a simplicial decomposition of a closed orientable surface. A triangulation is realizable in $\mathbb{R}^3$ when its vertices can be embedded in $\mathbb{R}^3$ so that its edges are straight, its triangles are flat, and distinct simplices do not intersect except along shared faces.

Geometric realization conjecture. Every triangulation of a closed orientable surface of genus at most $4$ is realizable in $\mathbb{R}^3$ by straight edges, flat triangles, and without self-intersections.

The conjecture is true for spheres, but its natural extension to closed orientable surfaces of greater genus is false. The stated genus-at-most-four claim is therefore resolved as part of the known geometric realization results for triangulated surfaces.