Regularity of minimal hypersurfaces with a common free boundary

Let $N$ be a Riemannian manifold and consider a stationary union of three or more $C^{1,\mu}$ hypersurfaces-with-boundary $M_k$ in $N$ with a common boundary $\Gamma$. We show that if $N$ is smooth, then $\Gamma$ is smooth and each $M_k$ is smooth up to $\Gamma$ (real analytic in the case $N$ is real analytic). Consequently we strengthen a result of Wickramasekera to conclude that under the stronger hypothesis that $V$ is a stationary, stable, integral $n$-varifold in an $(n+1)$-dimensional, smooth (real analytic) Riemannian manifold such that the support of $\|V\|$ is nowhere locally the union of three or more smooth (real analytic) hypersurfaces-with-boundary meeting along a common boundary, the singular set of $V$ is empty if $n = 6$, is discrete if $n = 7$, and has Hausdorff dimension at most $n-7$ if $n \geq 8$.