The Cantor-set conjecture for end invariants of one-holed torus characters

Let $\rho$ be an $\mathrm{SL}(2,\mathbb C)$ character, let $\mathcal E(\rho)$ be its set of end invariants, and let $\mathscr{PL}$ denote the projective lamination space. Cantor-set conjecture. If

$$\lvert\mathcal E(\rho)\rvert>2,$$

then either

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

or $\mathcal E(\rho)$ is a Cantor subset of $\mathscr{PL}$. This refines Bowditch's suggestion that, generically in $\mathcal X_{-2}$ outside the BQ-conditions, the end-invariant set is a Cantor set; the paper presents its results as evidence, but gives no resolution of the conjecture.