Comeager hereditary families of compact sets are big

Let $X$ be a Polish space and let $\mathcal K(X)$ be its Vietoris hyperspace. A family $\mathcal I\subseteq\mathcal K(X)$ is hereditary if it is downward closed under inclusion. Matheron and Zelený asked whether every comeager hereditary family in $\mathcal K(X)$ contains a dense hereditary $G_δ$ subfamily. We give an affirmative answer in ZFC.