Face-distance conjecture for torus decompositions of cubic irrationality

A torus decomposition arising from a periodic two-dimensional continued fraction of cubic irrationality has faces, each equipped with an integer distance to the origin. Face-distance conjecture. For any torus decomposition for the continued fraction of the cubic irrationality there exists some face with integer distance to the origin more than one. The conjecture is motivated by the failure of the simplest torus triangulation to arise from such a sail and by computations reported in the source; no resolution is given.