Determine the sharp weighted exponent for the strong maximal function

Determine the sharp exponent of the strong $A_p(\mathcal{R})$ characteristic governing the weighted $L^p(w)$ boundedness of the strong maximal function on $\mathbb{R}^d$ for $1<p<\infty$.

Background

The paper studies weighted bounds for the strong maximal function over axis-parallel rectangles in $\mathbb{R}^d$. A standard iteration of one-dimensional maximal operators gives an $A_p(\mathcal{R})$ exponent of $d/(p-1)$, while the paper proves improved upper bounds by exploiting the geometry of intersections of antichains of dyadic rectangles. The authors explicitly state that, despite the necessity of the strong $A_p(\mathcal{R})$ condition, the sharp exponent in the weighted bound remains unresolved. Their results therefore improve the best previously known upper exponent but do not identify the optimal one.