The Neoplatonic conjecture for Euclidean realizations

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. For a $6$-net $\sigma$, write $\Delta(\sigma)$ for its Euclidean realization, when it exists.

Neoplatonic conjecture. Any $6$-net $\sigma$ has a realization, unique up to isometry, as an undented Euclidean polyhedron $\Delta(\sigma)$ built from equilateral triangles of side length $1$.

The conjecture is motivated by computational experiments and by the proved existence results for all $6$-nets with at most $50$ vertices; uniqueness of the Euclidean realization is not established in the source.