  We show that the limit at infinity of the vector-valued Brown-York-type
quasi-local mass along any coordinate exhaustion of an asymptotically
hyperbolic $3$-manifold satisfying the relevant energy condition on the scalar
curvature has the conjectured causal character. Our proof uses spinors and
relies on a Witten-type formula expressing the asymptotic limit of this
quasi-local mass as a bulk integral which manifestly has the right sign under
the above assumptions. In the spirit of recent work by Hijazi, Montiel and
Raulot, we also provide another proof of this result which uses the theory of
boundary value problems for Dirac operators on compact domains to show that a
certain quasi-local mass, which converges to the Brown-York mass in the
asymptotic limit, has the expected causal character under suitable geometric
assumptions.
