The Nikodym maximal-function conjecture

For $1\leq d<n$, let $T_\delta(\sigma)$ be a $\delta$-plate oriented along $\sigma\in\mathrm{Gr}(d,n)$, and define the Nikodym maximal function

$$\mathcal N_\delta f(x)=\sup_{\sigma\in\mathrm{Gr}(d,n)}\fint_{x+T_\delta(\sigma)}|f|.$$

Nikodym maximal-function conjecture. For $1<p<n-d+1$,

$$\|\mathcal N_\delta:L^p(\mathbb{R}^n)\to L^p(\mathbb{R}^n)\|\lesssim \delta^{-\frac{n-d+1-p}{p}},$$

and for $p\geq n-d+1$,

$$\|\mathcal N_\delta:L^p(\mathbb{R}^n)\to L^p(\mathbb{R}^n)\|\lesssim (\log\delta^{-1})^{\frac1p}.$$

The conjecture is proposed as the sharp range of boundedness. The source says it is verified when $d=n-1$, where the critical exponent is $2$; the general statement remains open in the supplied text.