Linear decay conjecture for the transformed gradient of p-harmonic maps

Let $d,n\in\mathbb{N}$ and $1<p<\infty$. Let $h\colon\Omega\to\mathbb{R}^n$ be $p$-harmonic on an open set $\Omega\subset\mathbb{R}^d$, and let $V$ denote the standard nonlinear transformation associated with the $p$-Laplace equation. For a ball $B\subset\Omega$, write $\dashint_B$ for the average over $B$ and $\mean{V(\nabla h)}_B$ for the average of $V(\nabla h)$ over $B$. Linear decay conjecture. One has $V(\nabla h)\in C^1(\Omega)$ and there exists $c>0$ such that

$$\left(\dashint_{\theta B}\left|V(\nabla h)-\mean{V(\nabla h)}_{\theta B}\right|^2\,dx\right)^{\frac12}\leq c\,\theta\left(\dashint_B\left|V(\nabla h)-\mean{V(\nabla h)}_B\right|^2\,dx\right)^{\frac12}$$

for every ball $B\subset\Omega$ and every $\theta\in(0,1]$. The conjecture proposes the natural $C^1$ regularity and linear oscillation decay for $V(\nabla h)$ in arbitrary dimensions and for vectorial solutions; the preceding discussion notes that the corresponding linear behavior is not obtained from the available estimates, while planar regularity results motivate it. Its general validity remains open.