The prescribed metric on the boundary of convex subsets of anti-de Sitter space with a quasi-circle as ideal boundary

Let $h^{+}$ and $h^{-}$ be two complete, conformal metrics on the disc
$\mathbb{D}$. Assume moreover that the derivatives of the conformal factors of
the metrics $h^{+}$ and $h^{-}$ are bounded at any order with respect to the
hyperbolic metric, and that the metrics have curvatures in the interval
$\left(-\frac{1}{\epsilon}, -1 - \epsilon\right)$, for some $\epsilon > 0$. Let
$f$ be a quasi-symmetric map. We show the existence of a globally hyperbolic
convex subset $\Omega$ (see Definition 3.1) of the three-dimensional anti-de
Sitter space, such that $\Omega$ has $h^{+}$ (respectively $h^{-}$) as the
induced metric on its future boundary (respectively on its past boundary) and
has a gluing map $\Phi_{\Omega}$ (see Definition 4.5) equal to $f$.