Conjecture on regularity of $J_p$-harmonic maps

Let $M$ and $N$ be hyperbolic surfaces, let $p>2$, and let $u_p:M\rightarrow N$ satisfy the $J_p$-Euler–Lagrange equations. Regularity conjecture. The map $u_p$ belongs to $C^{1,\alpha}$. The paper notes that minimizing $p$-harmonic maps have this regularity, while the corresponding higher regularity for $J_p$-minimizers is left as an open question.