The off-diagonal Nikodym maximal-function conjecture

For $1\leq d<n$, let $\mathcal N_\delta$ denote the $(d,n)$-Nikodym maximal function. For $1\leq p\leq n-d+1$, set $q=p'(n-d)$, where $p'$ is the Hölder conjugate of $p$. Off-diagonal Nikodym conjecture. For all $\varepsilon>0$,

$$\|\mathcal N_\delta f:L^p(\mathbb{R}^n)\to L^q(\mathbb{R}^n)\|\lesssim_\varepsilon \delta^{-\left(\frac{n-d+1-p}{p}\right)}(\log\delta^{-1})^{\frac{1}{p'(n-d)}}.$$

This is proposed by interpolating the preceding conjectural estimate with the trivial $L^1\to L^\infty$ bound. The supplied text gives no resolution of this off-diagonal conjecture.