The $C^{p^{\mathrm{\prime }}}$-regularity conjecture near $p=2$

We prove the $C^{p^{\mathrm{\prime }}}$-regularity conjecture in every dimension when $p>2$ is sufficiently close to $2$. To this end, we establish improved Hölder estimates for the gradients of $p$-harmonic functions. These estimates also determine the first-order asymptotics of the optimal gradient Hölder exponent in every dimension. The proof combines compactness, harmonic rigidity of the limiting profiles, and a sharp uniform gap estimate for the first variation of the gradient excess.