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We consider integral area-minimizing $2$-dimensional currents $T$ in$U\subset \mathbb R^{2+n}$ with $\partial T = Q[\![\Gamma]\!]$, where $Q\in\mathbb N \setminus \{0\}$ and $\Gamma$ is sufficiently smooth. We prove that,if $q\in \Gamma$ is a point where the density of $T$ is strictly below$\frac{Q+1}{2}$, then the current is regular at $q$. The regularity isunderstood in the following sense: there is a neighborhood of $q$ in which $T$consists of a finite number of regular minimal submanifolds meetingtransversally at $\Gamma$ (and counted with the appropriate integermultiplicity). In view of well-known examples, our result is optimal, and it isthe first nontrivial generalization of a classical theorem of Allard for $Q=1$.As a corollary, if $\Omega\subset \mathbb R^{2+n}$ is a bounded uniformlyconvex set and $\Gamma\subset \partial \Omega$ a smooth $1$-dimensional closedsubmanifold, then any area-minimizing current $T$ with $\partial T = Q[\![\Gamma]\!]$ is regular in a neighborhood of $\Gamma$.

An Allard-type boundary regularity theorem for $2d$ minimizing currents at smooth curves with arbitrary multiplicity