  We consider bipolar oriented random planar maps with heavy-tailed face
degrees. We show for each $\alpha \in (1,2)$ that if the face degree is in the
domain of attraction of an $\alpha$-stable L\'evy process, the corresponding
random planar map has an infinite volume limit in the Benjamini-Schramm
topology. We also show in the limit that the properly rescaled contour
functions associated with the northwest and southeast trees converge in law to
a certain correlated pair of $\alpha$-stable L\'evy processes. Combined with
other work, this allows us to identify the scaling limit of the planar map with
an SLE$_\kappa(\rho)$ process with $\rho = \kappa-4 < -2$ on
$\sqrt{\kappa}$-Liouville quantum gravity for $\kappa \in (4/3,2)$ where
$\alpha, \kappa$ are related by $\alpha = 4/\kappa-1$.
