It is well-known that continuous weakly harmonic maps are smooth. Is the same thing true for $p$-harmonic maps (for $p>2$)?
More specifically: Let $M,N$ be smooth Riemannian manifolds, and let $f \in W^{1,p}(M,N)$ be a continuous map, which is also weakly $p$-harmonic. Is $f$ smooth?
I am in particularly interested in the case where the source and target have the same dimension $d$, and $p=d$.
I am somewhat aware that the regularity theory for $p$-harmonic maps is less developed in general compared to the classic $p=2$ case, but I could not find a specific reference discussing this specific question.