Weighted Musielak–Orlicz maximal-operator boundedness conjecture

Let $\phi$ be a generalized $\Phi$-function, let $\phi^*$ be its complementary function, and define

$$\phi_w(x,t):=\phi(x,w(x)t).$$

Assume that $\phi^*$ satisfies the $\Delta_2$ condition and that $\phi$ satisfies conditions $(A0)$–$(A2)$. Let $w$ be a weight such that $L^{\phi_w(\cdot)}({\mathbf R}^d)\in A$. Weighted Musielak–Orlicz boundedness conjecture. Then

$$M:L^{\phi_w(\cdot)}({\mathbf R}^d)\to L^{\phi_w(\cdot)}({\mathbf R}^d).$$

This proposes a weighted extension of known maximal-operator boundedness results for Musielak–Orlicz spaces. The source gives no resolution.