Local Hölder regularity conjecture for homogeneous fractional p-Laplace systems

Let $n,N\in\mathbb{N}$ and let $\Omega$ be a bounded open set in $\mathbb{R}^n$. For $s\in(0,1)$ and $p\in(1,\infty)$, let $u\in W^{s,p}_{\mathrm{loc}}(\Omega,\mathbb{R}^N)$ satisfy

$$\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(\varphi(x)-\varphi(y))}{|x-y|^{n+sp}}\,dx\,dy=0\qquad\forall\varphi\in C_c^\infty(\Omega,\mathbb{R}^N).$$

Local Hölder regularity conjecture. Then $u$ is locally Hölder continuous in $\Omega$.

This is the regularity statement for homogeneous fractional $p$-Laplace systems needed to remove the paper's restriction that $sp$ lie sufficiently close to $n$. It is presented as an unknown result and is used as the missing ingredient in a blow-up argument for regularity of minimizing fractional harmonic maps.