Necessary conditions for the existence of wandering triangles for cubic laminations

In his 84 preprint W. Thurston proved that quadratic laminations do not admit so-called wandering triangles and asked a deep question concerning their existence for laminations of higher degrees. Recently it has been discovered by L. Oversteegen and the author that some closed laminations of the unit circle invariant under $z\mapsto z^d, d>2$ admit wandering triangles. This makes the problem of describing the criteria for the existence of wandering triangles important because solving this problem would help understand the combinatorial structure of the family of all polynomials of the appropriate degree.
In this paper for a closed lamination on the unit circle invariant under $z\mapsto z^3$ (cubic lamination) we prove that if it has a wandering triangle then there must be two distinct recurrent critical points in the corresponding quotient space ("topological Julia set") $J$ with the same limit set coinciding with the limit set of any wandering vertex (wandering vertices in $J$ correspond to wandering gaps in the lamination).