  In this paper we show that, if $T$ is an area-minimizing $2$-dimensional
integral current with $\partial T = Q [\![ \Gamma ]\!]$, where $\Gamma$ is a
$C^{1,\alpha}$ curve for $\alpha>0$ and $Q$ an arbitrary integer, then $T$ has
a unique tangent cone at every boundary point, with a polynomial convergence
rate. The proof is a simple reduction to the case $Q=1$, studied by Hirsch and
Marini.
