Unit-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 with integer distances to the origin. Unit-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 equals one. The source presents this as an observation from calculated sails, while giving no proof or resolution.