Global strong Muckenhoupt characterization for weighted Morrey spaces

Let $1<p\leq q<\infty$, let $w$ be a weight, and let $M$ denote the Hardy–Littlewood maximal operator. Let $T$ range over all Calderón–Zygmund operators and let $R_j$, $j=1,\ldots,d$, denote the Riesz transforms. Global weighted Morrey characterization conjecture. The following are equivalent:

1. $T:M_w^{p,q}({\mathbf R}^d)\to M_w^{p,q}({\mathbf R}^d)$ for every Calderón–Zygmund operator $T$;
2. $R_j:M_w^{p,q}({\mathbf R}^d)\to M_w^{p,q}({\mathbf R}^d)$ for every $j=1,\ldots,d$;
3. Both

$$M:M_w^{p,q}({\mathbf R}^d)\to M_w^{p,q}({\mathbf R}^d)$$

and

$$M:B_{w^{-1}}^{p',q'}({\mathbf R}^d)\to B_{w^{-1}}^{p',q'}({\mathbf R}^d);$$

4. $M_w^{p,q}({\mathbf R}^d)\in A_{\mathrm{strong}}$.

This conjecture globalizes a local characterization of weighted Morrey spaces, replacing the centered-at-zero cube collection by all cubes and requiring the stronger Muckenhoupt condition. The source presents it as a conjecture and gives no resolution.