The ideal neoplatonic conjecture

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 whose ideal triangular faces meet midpoint-to-midpoint, where the midpoint of an infinite edge is the foot of the perpendicular dropped from the opposite vertex. For a $6$-net $\sigma$, write $\Delta(\sigma,\infty)$ for its ideal realization.

Ideal neoplatonic conjecture. Every $6$-net $\sigma$ has a realization $\Delta(\sigma,\infty)$, unique up to isometry, as an ideal neoplatonic.

The conjecture proposes existence and uniqueness for all $6$-nets. The source proves a convex ideal neoplatonic realization, unique up to isometry, for every prime $6$-net with at most $50$ vertices, but does not establish the general assertion.