  The mirror dual of a smooth toric Fano surface $X$ equipped with an
anticanonical divisor $E$ is a Landau-Ginzburg model with superpotential, W.
Carl-Pumperla-Siebert give a definition of the the superpotential in terms of
tropical disks using a toric degeneration of the pair $(X,E)$. When $E$ is
smooth, the superpotential is proper. We show that this proper superpotential
equals the open mirror map for outer Aganagic-Vafa branes in the canonical
bundle $K_X$, in framing zero. As a consequence, the proper Landau-Ginzburg
potential is a solution to the Lerche-Mayr Picard-Fuchs equation.
  Along the way, we prove a generalization of a result about relative
Gromov-Witten invariants by Cadman-Chen to arbitrary genus using the
multiplication rule of quantum theta functions. In addition, we generalize a
theorem of Hu that relates Gromov-Witten invariants of a surface under a
blow-up from the absolute to the relative case. One of the two proofs that we
give introduces birational modifications of a scattering diagram. We also
demonstrate how the Hori-Vafa superpotential is related to the proper
superpotential by mutations from a toric chamber to the unbounded chamber of
the scattering diagram.
