Thurston's convex-core bending-lamination conjecture

Let $S$ be a connected closed surface and let $\mathcal{MGL}_{\pi}(S)$ denote the measured geodesic laminations on $S$ whose closed leaves have weight strictly smaller than $\pi$. Say that two measured laminations $L^{+}$ and $L^{-}$ fill $S$ if there exists $\epsilon>0$ such that, for every simple closed curve $\alpha$,

$$i(\alpha,L^{+})+i(\alpha,L^{-})>\epsilon.$$

For a non-Fuchsian quasi-Fuchsian manifold $Q$, write $\partial^{\pm}C(Q)$ for the two convex-core boundary components and $L^{\pm}$ for their measured bending laminations.

Thurston's conjecture. Given any pair of measured laminations $L^{+},L^{-}\in\mathcal{MGL}_{\pi}(S)$ that fill $S$, there is a unique quasi-Fuchsian manifold $Q$, up to isotopy, such that $L^{+}$ is the bending lamination of $\partial^{+}C(Q)$ and $L^{-}$ is the bending lamination of $\partial^{-}C(Q)$.

Bonahon and Otal established the existence part, but uniqueness is known only in particular cases and remains open in general. Equivalently, the bending-lamination map should be bijective onto the space of filling pairs of measured laminations with closed-leaf weights below $\pi$.