The arithmetic lacunary spherical maximal weak-type $(1,1)$ conjecture

Let $d\geq5$, let $\mathcal{R}$ be a lacunary subsequence of $\mathcal{R}_{full}$, and let $A_{\mathcal{R}}^*$ be the associated discrete spherical maximal operator. Weak-type $(1,1)$ conjecture. One has

$$A_{\mathcal{R}}^*:\ell^1(\mathbb{Z}^d)\to\ell^{1,\infty}(\mathbb{Z}^d).$$

The paper identifies this as its motivation, but later notes that the unrestricted version fails in general because of an arithmetic obstruction.