Let $M$ be a smooth, bounded, oriented Riemannian manifold-with-boundary. Let $\alpha$ be a harmonic differential $p$-form on $M$, subject to the boundary condition $\alpha\wedge\nu^\sharp|\partial M = 0$ or $\iota_\nu \alpha|\partial M = 0$. Here $\nu$ is the normal vectorfield along $\partial M$, $\nu^\sharp$ is its dual $1$-form, and $\iota$ is the interior multiplication. Assume that $\alpha \in W^{1,2}$.

The question is: can we conclude that $\alpha \in C^\infty$?