Soft polyhedric realization conjecture for balanced normal convex tilings

Let $M$ be a balanced, normal, convex tiling.

Soft polyhedric realization conjecture. There exists a combinatorially equivalent, soft polyhedric tiling $M'$.

Theorem 1 proves this conclusion under the stronger assumption that every dual of a vertex polyhedron in $M$ has a Hamiltonian circuit. The source suggests that this sufficient condition may be relaxable, but does not establish the conjecture in general.