Magyar–Stein–Wainger maximal inequality in four dimensions

For $f:\mathbb{Z}^4\to\mathbb{C}$, define the discrete spherical maximal function by

$$A_*f({\bf x}):=\sup_{\lambda\in\mathbb{N}:\,\lambda\ \text{is odd}}|A_\lambda f({\bf x})|.$$

Magyar–Stein–Wainger conjecture. If $p>2$, then

$$\|A_* f\|_{\ell^{p}(\mathbb{Z}^4)}\lesssim_p\|f\|_{\ell^{p}(\mathbb{Z}^4)}.$$

This is the expected four-dimensional extension of the discrete spherical maximal theorem, which is known in dimensions at least five but remains open in dimension four.