The ideal-prime conjecture for convexity

A 6-net is a simplicial triangulation of the $2$-sphere with maximum degree at most $6$. A net is prime if every $3$-cycle bounds a face. An ideal neoplatonic is an equilateral ideal hyperbolic polyhedron, and $\Delta(\sigma,\infty)$ denotes the ideal neoplatonic realization associated to a $6$-net $\sigma$.

Ideal-prime conjecture. If the $6$-net $\sigma$ is prime, its ideal neoplatonic realization $\Delta(\sigma,\infty)$ is convex.

The source proves this assertion for prime $6$-nets with $v\leq50$, including uniqueness of the convex realization. Its validity for arbitrary prime $6$-nets remains open in the supplied text.