Strong spherical maximal function conjecture

Let $n\geq 2$. For a tuple of dilation parameters $\mathbf{r}=(r_1,\dots,r_n)\in(0,\infty)^n$, define

$$\boldsymbol{\delta}_{\mathbf{r}}(x_1,\dots,x_n)=(r_1x_1,\dots,r_nx_n).$$

Let $\sigma$ be normalized surface measure on the unit sphere $S^{n-1}$, and define its dilation $\sigma_{\mathbf{r}}$ by

$$\langle\sigma_{\mathbf{r}},f\rangle=\langle\sigma,f\circ\boldsymbol{\delta}_{\mathbf{r}}\rangle.$$

The strong spherical maximal function is

$$\mathcal{M}_{\mathrm{st}}f(x)=\sup_{\mathbf{r}\in(0,\infty)^n}|f*\sigma_{\mathbf{r}}(x)|.$$

Strong spherical maximal function conjecture. For all $n\geq 2$ and $p>\frac{n+1}{n-1}$, there exists a constant $C_{n,p}>0$ such that

$$\|\mathcal{M}_{\mathrm{st}}f\|_{L^p(\mathbb{R}^n)}\leq C_{n,p}\|f\|_{L^p(\mathbb{R}^n)}$$

for all $f\in C_c(\mathbb{R}^n)$. This would give the conjecturally sharp $L^p$ range for the strong spherical maximal operator; the paper proves the estimate for all $n\geq3$ and $p>2$, which is sharp when $n=3$, while the conjectured range remains open in higher dimensions.