The dendrite conjecture for end invariants of one-holed torus characters

Let $\rho$ be an $\mathrm{SL}(2,\mathbb C)$ character and let ${{\mathcal E}(\rho)}$ denote its set of end invariants, viewed as a subset of the projective lamination space ${{\mathscr {PL}}}$. The dendrite conjecture. If ${{\mathcal E}(\rho)}$ has more than two elements, then either

$${{\mathcal E}(\rho)}={{\mathscr {PL}}}$$

or ${{\mathcal E}(\rho)}$ is a Cantor subset of ${{\mathscr {PL}}}$. This refines Bowditch's suggestion for generic type-preserving characters not satisfying the BQ-conditions; the preceding results provide supporting evidence, and the associated convex-hull tree is expected to have dendrite-like branching.