  Motivated by Thurston and Daskalopoulos--Uhlenbeck's approach to
Teichm\"uller theory, we study the behavior of $q$-harmonic functions and their
$p$-harmonic conjugates in the limit as $q \to 1$, where $1/p + 1/q = 1$. The
$1$-Laplacian is already known to give rise to laminations by minimal
hypersurfaces; we show that the limiting $p$-harmonic conjugates converge to
calibrations $F$ of the laminations. Moreover, we show that the laminations
which are calibrated by $F$ are exactly those which arise from the
$1$-Laplacian. We also explore the limiting dual problem as a model problem for
the optimal Lipschitz extension problem, which exhibits behavior rather unlike
the scalar $\infty$-Laplacian. In a companion work, we will apply the main
result of this paper to associate to each class in $H^{d - 1}$ a lamination in
a canonical way, and study the duality of the stable norm on $H_{d - 1}$.
