Maximal-operator duality for weighted Morrey and block spaces

Let $1<p\leq q<\infty$, let $w$ be a weight, and let $M$ denote the Hardy–Littlewood maximal operator. Define the weighted Morrey and block spaces by

$$\|f\|_{M_w^{p,q}({\mathbf R}^d)}:=\|fw\|_{M^{p,q}({\mathbf R}^d)},\qquad \|g\|_{B_{w^{-1}}^{p',q'}({\mathbf R}^d)}:=\|gw^{-1}\|_{B^{p',q'}({\mathbf R}^d)}.$$

Morrey–block duality conjecture. If

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

then

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

This is the Morrey-space form of the one-sided maximal-operator duality question, using the Köthe duality between weighted Morrey and block spaces. The source explicitly states that the converse implication is false, so the asserted implication itself remains unresolved.