  We prove a conjecture of Benjamini and Curien stating that the local limits
of uniform random triangulations whose genus is proportional to the number of
faces are the Planar Stochastic Hyperbolic Triangulations (PSHT) defined in
arXiv:1401.3297. The proof relies on a combinatorial argument and the
Goulden--Jackson recurrence relation to obtain tightness, and probabilistic
arguments showing the uniqueness of the limit. As a consequence, we obtain
asymptotics up to subexponential factors on the number of triangulations when
both the size and the genus go to infinity.
  As a part of our proof, we also obtain the following result of independent
interest: if a random triangulation of the plane $T$ is weakly Markovian in the
sense that the probability to observe a finite triangulation $t$ around the
root only depends on the perimeter and volume of $t$, then $T$ is a mixture of
PSHT.
