Thurston-algorithm ideal-boundary conjecture

Let an admissible combinatorics be given, and consider the successive conjugacy classes generated by the Thurston algorithm. Let the shift locus denote the locus of conjugacy classes with the relevant expanding dynamical behavior, and let $(\Sigma,\Delta)$ be the moduli coordinates. Thurston-algorithm ideal-boundary conjecture. For any admissible combinatorics, either at most finitely many of the successive conjugacy classes generated by the Thurston algorithm belong to the shift locus, or the successive conjugacy classes converge to one of the two ideal points with coordinates $(\Sigma,\Delta)=(\pm.5,1)$, or both conditions hold. This conjecture is intended to describe the alternatives governing convergence of the Thurston algorithm; the source provides examples and discussion but no proof of the general assertion.