Triangular-face conjecture for torus decompositions of cubic irrationality

A torus decomposition arising from a periodic two-dimensional continued fraction of cubic irrationality consists of faces with integer affine types. Triangular-face conjecture. For any torus decomposition for the continued fraction of the cubic irrationality there exists the face that has an integer affine type of some triangle. The conjecture rules out torus decompositions all of whose faces have nontriangular integer affine types, including the one-rectangular-face example discussed immediately before it; no resolution is given.