The Neoplatonic homotopy conjecture

A 6-net is a simplicial triangulation of the $2$-sphere with maximum degree at most $6$. For a $6$-net $\sigma$ and $0<l\leq\infty$, let $\Delta(\sigma,l)$ denote a realization at side length $l$. An undented hyperbolic polyhedron is a hyperbolic polyhedron without dents; $\Delta(\sigma,\infty)$ is the ideal endpoint, and $\Delta(\sigma)$ is the Euclidean neoplatonic realization.

Neoplatonic homotopy conjecture. Every $6$-net $\sigma$ has a family of realizations $\Delta(\sigma,l)$, each unique up to isometry, as undented hyperbolic polyhedra of side length $l$, $0<l\leq\infty$, where $\Delta(\sigma,\infty)$ is an equilateral ideal hyperbolic polyhedron. After rescaling all lengths by $1/l$, $\Delta(\sigma,l)$ converges as $l\rightarrow0$ to the Euclidean neoplatonic realization $\Delta(\sigma)$.

The conjecture describes a continuous deformation from the ideal hyperbolic realization to the Euclidean one while preserving undentedness. The source gives animations illustrating this deformation but does not prove the asserted family or convergence.