Stein's dimension-free maximal inequality conjecture for convex symmetric bodies

Let $d\in\mathbb{Z}_+$, let $\mathfrak B(d)$ be the set of all convex symmetric bodies in $\mathbb{R}^d$, and for $U\in\mathfrak B(d)$ let $\mathcal C(p,\mathbb{R}_+,U)$ be the smallest constant in the discrete maximal inequality

$$\left\|\sup_{t>0}|\mathcal M_t^U f|\right\|_{\ell^p(\mathbb{Z}^d)}\leq \mathcal C(p,\mathbb{R}_+,U)\|f\|_{\ell^p(\mathbb{Z}^d)}.$$

Stein's conjecture. For every $p\in(1,\infty]$ there is a constant $C_p>0$ such that

$$\sup_{d\in\mathbb{Z}_+}\sup_{U\in\mathfrak B(d)}\mathcal C(p,\mathbb{R}_+,U)\leq C_p.$$

This asks for dimension-free bounds for discrete maximal operators associated with convex symmetric bodies. The corresponding bound is finite in each fixed dimension for every $p\in(1,\infty]$, but uniformity over all dimensions and all convex symmetric bodies remains a major open problem arising from Stein's work.