Hardy-Littlewood maximal operator on the associate space of a Banach function space

Let $\mathcal{E}(X,d,\mu)$ be a Banach function space over a space of homogeneous type $(X,d,\mu)$. We show that if the Hardy-Littlewood maximal operator $M$ is bounded on the space $\mathcal{E}(X,d,\mu)$, then its boundedness on the associate space $\mathcal{E}'(X,d,\mu)$ is equivalent to a certain condition $\mathcal{A}_\infty$. This result extends a theorem by Andrei Lerner from the Euclidean setting of $\mathbb{R}^n$ to the setting of spaces of homogeneous type.