Non-vanishing conjecture for homogeneous polynomial solutions of the p-Laplace equation

Let $u$ be a real homogeneous polynomial of degree $m=\deg u\ge 2$ on $\mathbb{R}^n$, with $n\ge 3$. Define the $p$-Laplace operator by

$$\Delta_p u:=|D u|^2\Delta u+\frac{p-2}{2}\langle D u,D|D u|^2\rangle.$$

Non-vanishing conjecture. If $\Delta_pu=0$ for $p>1$ and $p\ne2$, then $u\equiv 0$.

This conjecture is equivalent to the non-vanishing property for real analytic solutions of the $p$-Laplace equation in $\mathbb{R}^n$ for $n\ge3$: a real analytic solution with a vanishing gradient at an interior point should be identically zero. The corresponding statement in dimension two was proved by John L. Lewis, while the general case remains open in the supplied source.