The volume-maximizing conjecture for prime 6-nets

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 geodesic $2$-cycle with combinatorics $\sigma$ is an oriented ideal geodesic realization of the triangulation $\sigma$, and its volume is interpreted algebraically. Let the ideal neoplatonic realization of a prime $6$-net $\sigma$ be the convex ideal realization described above.

Volume-max conjecture. For a prime $6$-net $\sigma$, its ideal neoplatonic realization maximizes volume among all ideal geodesic $2$-cycles with combinatorics $\sigma$.

Rivin's result gives the corresponding maximization among convex ideal realizations. The conjecture asks whether convexity can be dropped; the source notes that the non-prime case follows by splitting along separating triangles.