Maximal-operator duality for Musielak–Orlicz spaces

Let $\phi$ be a generalized $\Phi$-function, let $\phi^*$ be its complementary generalized $\Phi$-function, and let $M$ denote the Hardy–Littlewood maximal operator. Assume that both $\phi$ and $\phi^*$ satisfy the $\Delta_2$ condition. Musielak–Orlicz duality conjecture. Then

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

if and only if

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

This is the Musielak–Orlicz specialization of the convex-concave Banach-function-space duality conjecture. The source provides no resolution.