Hougardy–Lutz–Zelke realizability conjecture for triangulated orientable surfaces

An orientable surface triangulation is a triangulation of an orientable surface of genus $g$, where $g$ is a nonnegative integer. A triangulation is geometrically realizable if it can be realized in ${\mathbb R}^3$ with straight edges, flat triangles, and without self-intersections.

Hougardy–Lutz–Zelke conjecture. Every triangulation of an orientable surface of genus $g$ with

$$1\leq g\leq 4$$

is geometrically realizable.

The claim is motivated by the computational realizability of all vertex-minimal triangulations in genera $2$, $3$, and $4$, while non-realizable triangulations are known for every orientable genus $g\geq 5$.