  We revisit the 3-states Potts model on random planar triangulations as a
Hermitian matrix model. As a novelty, we obtain an algebraic curve which
encodes the partition function on the disc with both fixed and mixed spin
boundary conditions. We investigate the critical behaviour of this model and
find scaling exponents consistent with previous literature. We argue that the
conformal field theory that describes the double scaling limit is Liouville
quantum gravity coupled to the $(A_4,D_4)$ minimal model with extended
$\mathcal{W}_3$-symmetry.
