Existence and uniqueness conjecture for AdS manifolds with prescribed convex boundary metrics

Let $S$ be the surface in question, and let $h_+$ and $h_-$ be smooth metrics on $S$ whose curvature is $K<-1$. A globally hyperbolic $\mathbb{A}\mathbb{d}\mathbb{S}^3$ manifold is a spacetime homeomorphic to $S\times[-1,1]$; its boundary is smooth, space-like and strictly convex when it has these properties, and the induced metrics on the two boundary components are prescribed by $h_-$ and $h_+$.

AdS convex-boundary conjecture. There exists a unique globally hyperbolic $\mathbb{A}\mathbb{d}\mathbb{S}^3$ manifold homeomorphic to $S\times[-1,1]$, with smooth, space-like and strictly convex boundary, such that the induced metric on $S\times\\{-1\\}$ is $h_-$ and the induced metric on $S\times\\{1\\}$ is $h_+$.

The constant-curvature case supplies the existence part by the theorem stated immediately before the conjecture. The analogous assertion for quasifuchsian, and more generally convex co-compact, hyperbolic manifolds is known; the general AdS statement remains open.