  We prove that, for a $C^2$ partially hyperbolic endomorphism of the 2-torus
which is strongly transitive, given an ergodic $u$-Gibbs measure that has
positive center Lyapunov exponent and has full support, then either the map is
special (has only one unstable direction per point), or the measure is the
unique absolutely continuous invariant measure. We can apply this result in
many settings, in particular obtaining uniqueness of $u$-Gibbs measures for
every non-special perturbation of irreducible linear expanding maps of the
torus with simple spectrum. This gives new open sets of partially hyperbolic
systems displaying a unique $u$-Gibbs measure.
