Chvátal's conjecture on subset-closed families

Let ${\cal F}$ be a family of sets. It is subset-closed if every subset of every member of ${\cal F}$ also belongs to ${\cal F}$. Recall that ${\cal F}$ is ${\sf EKR}$ when some star ${\cal F}_x$ has size at least that of every intersecting subfamily ${\cal H}\subseteq{\cal F}$.

Chvátal's conjecture. Every subset-closed family of sets is ${\sf EKR}$.

This conjecture asks for an Erdős–Ko–Rado-type extremal theorem for arbitrary subset-closed families. Its status is not specified in the source.