Non-meager $\mathsf{P}$-filters, Miller-measurability, and a question of Hrušák

Given a cardinal $\kappa$ and filters ${\mathcal{F}}_{\alpha }$ on $\omega$ for $\alpha \in \kappa$, we will show that if ${\prod }_{\alpha \in \kappa }{\mathcal{F}}_{\alpha }$ is countable dense homogeneous then $\kappa <\mathfrak{p}$ and each ${\mathcal{F}}_{\alpha }$ is a non-meager $\mathsf{P}$-filter. This partially answers a question of Michael Hrušák. Along the way, we will show that the product of fewer than $\mathfrak{p}$ non-meager $\mathsf{P}$-filters has the Miller property. We will also describe explicitly the connection between Miller-measurability and the Miller property. As a corollary, we will see that the intersection of fewer than $\mathsf{a}\mathsf{d}\mathsf{d}(m^0)$ non-meager $\mathsf{P}$-filters is a non-meager $\mathsf{P}$-filter, where $m^0$ denotes the ideal of Miller-null sets. We will conclude by investigating the preservation of the Miller property under intersections and products.