Properly embedded Legendrian surfaces in the tangent sphere bundle of hyperbolic space

We show that every open Riemann surface may be conformally properly embedded in the tangent sphere bundle of hyperbolic $3$-space as a Legendrian surface. We achieve this by constructing proper holomorphic Legendrian embeddings in $\mathrm{SL}_2 (\mathbb{C})$ whose projections to the quotient $\mathrm{SL}_2 (\mathbb{C})/ \mathrm{U}_1$ remain embeddings. As another consequence, every open Riemann surface admits a proper holomorphic Legendrian embedding in the projectivised special linear group $\mathrm{PSL}_2 (\mathbb{C})$.