The four-facet conjecture for three-dimensional staircase Kunz semigroups

Let $N$ be a numerical semigroup with associated face $F_N$, and suppose that $N$ is staircase and Kunz. When $k=3$, the four-facet conjecture. $F_N$ has at most $4$ facets.

This conjecture concerns the number of facets of chambers arising from three-dimensional staircase Kunz semigroups. It is reported as having been verified computationally for $m\leq 42$, although the authors express some reservation; the broader related conjecture for arbitrary three-dimensional lattice ideals was counterexampled.