Uniqueness of the mountain-range diagram for non-loose torus knots

Let $(p,q)$ be a pair of coprime integers specifying a torus knot, and consider an overtwisted contact structure that supports non-loose Legendrian $(p,q)$--torus knots. The mountain range is the collection of these knots organized by their classical invariants and destabilization relations. Mountain-range uniqueness conjecture. In each overtwisted contact structure that supports non-loose $(p,q)$--torus knots, the mountain range of such knots is given by only one of the diagrams indicated in Figures~ and~. The conjecture asserts that the mountain ranges depicted in the generic and exceptional cases never occur together in a single overtwisted contact structure; it is based on the authors' computed examples, while no general proof is given.