Resolve the endpoint \(p=1\) multiplier weak-type characterization

Determine whether \(w\in A_{(1,1),(q,\infty)}\) is sufficient for boundedness of \(M_\alpha:L_w^{1,1}(\mathbb R^d)\to L_w^{q,\infty}(\mathbb R^d)\), where \(q=d/(d-\alpha)\).

Background

At the endpoint $p=1$, the paper proves that boundedness implies $w\in A_{(1,1),(q,\infty)}$, while the classical class $A_{1,q}$ is sufficient. The sufficiency of the larger Lorentz–Muckenhoupt class is unresolved; for $\alpha=0$, this specializes to the Muckenhoupt–Wheeden problem.