The $C^{p'}$-regularity conjecture for elliptic $p$-Laplace equations

Let $p\in(2,\infty)$, let $B_1$ be the unit ball, and let $\Delta_p u\coloneqq\operatorname{div}(|Du|^{p-2}Du)$. A function is locally of class $C^{1,\alpha}$ if it has locally Hölder-continuous first derivatives with exponent $\alpha$. $C^{p'}$-regularity conjecture. Every solution $u$ of

$$-\Delta_p u=f\in L^{\infty}(B_1)$$

is locally of class $C^{1,\frac{1}{p-1}}=C^{p'}$, where $p'=\frac{p}{p-1}$. This is a well-known open problem concerning optimal regularity in the elliptic setting; the paper studies related optimal regularity for parabolic $p$-Laplace equations.