Existence of good geometric triangulations for hyperbolic 3-manifolds with geodesic boundary

Let $N$ be a hyperbolic $3$-manifold with non-empty geodesic boundary. A partially truncated triangulation of $N$ is a realization of $N$ as a gluing of partially truncated tetrahedra along their lateral hexagons. It is good if every tetrahedron has at most one ideal vertex, and it is geometric if the hyperbolic structure on $N$ is obtained by gluing geometric realizations of the tetrahedra along isometries of their lateral hexagons.

Good geometric triangulation conjecture. Every hyperbolic $N$ with non-empty geodesic boundary admits a good geometric triangulation.

Kojima's canonical decomposition provides a geometric decomposition into partially truncated polyhedra, each with at most one ideal vertex, and many such decompositions can be subdivided into geometric triangulations. The conjecture asserts that this subdivision is always possible for a good triangulation.