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

Uniqueness of boundary tangent cones for $2$-dimensional area-minimizing currents