The three-dimensional anisotropic Kakeya--Nikodym conjecture

Let $n=3$, let $\Sigma$ be a strictly convex hypersurface, and let $\mathop{\mathrm{MKNE}}$ denote the maximal Kakeya--Nikodym estimate referred to in the source as equation (MKNE). Three-dimensional anisotropic Kakeya--Nikodym conjecture. The estimate $\mathop{\mathrm{MKNE}}$ should hold for every $p\ge3$.

The source explains that this conjecture would imply Stein's Fourier restriction conjecture in dimension three. Its status is not resolved in the supplied context.