Establish sufficiency of the Lorentz–Muckenhoupt condition above the target index

Prove or disprove that \(w\in A_{(p,r),(q,s)}\) is sufficient for boundedness of \(M_\alpha:L_w^{p,r}\to L_w^{q,s}\) when \(p>1\), \(p<r\le s\), and \(s>q\).

Background

For $p<r\le s$ with $s>q$, the paper establishes the Lorentz–Muckenhoupt condition as necessary for the weighted maximal-operator estimate. Unlike the case $r=p$, the paper does not establish a matching sufficiency theorem, leaving open whether the displayed condition gives the exact characterization.