Mess's uniqueness conjecture for maximal globally hyperbolic AdS manifolds

Let $S$ be the underlying surface, and let $h_-$ and $h_+$ be hyperbolic metrics on $S$. A globally hyperbolic $\mathbb{A}\mathbb{d}\mathbb{S}^3$ manifold is maximal when it is maximal among globally hyperbolic AdS structures, and its convex core has a boundary whose induced metrics are prescribed by $h_-$ and $h_+$.

Mess's conjecture. There is a unique maximal globally hyperbolic $\mathbb{A}\mathbb{d}\mathbb{S}^3$ manifold $N$ such that the induced metric on the boundary of the convex core of $N$ is given by $h_-$ and $h_+$.

The source presents this as the limiting case, as the curvatures tend to $-1$, of the preceding conjecture or of the constant-curvature theorem. It attributes the statement to Mess and gives no resolution, so it remains open here.