The ending invariant rigidity conjecture for one-holed torus characters

Let ${\mathcal X}_{\kappa}$ be the character variety with parameter $\kappa$, and let ${{\mathcal E}(\rho)}$ be the set of end invariants of a character $\rho$. The ending invariant rigidity conjecture. Suppose that $\rho,\rho'\in{\mathcal X}_{\kappa}$ satisfy

$${{\mathcal E}(\rho)}={{\mathcal E}(\rho')},\qquad |{{\mathcal E}(\rho)}|\geq 2,$$

and

$${{\mathcal E}(\rho)}\neq{{\mathscr {PL}}}.$$

Then $\rho=\rho'$. This is presented as a generalization of the Ending Lamination Conjecture for $\mathrm{SL}(2,\mathbb C)$ characters; its resolution is not indicated in the supplied text.