Chvátal's conjecture on hereditary families

Let ${\cal F}$ be a family of sets. Call ${\cal F}$ hereditary if every subset of every member of ${\cal F}$ also belongs to ${\cal F}$. Call ${\cal F}$ EKR if some element $x$ satisfies $|{\cal F}_x|\geq |{\cal H}|$ for every intersecting subfamily ${\cal H}\subseteq{\cal F}$. Chvátal's conjecture. Every hereditary family ${\cal F}$ is EKR. The paper describes this as a longstanding and difficult conjecture, with few papers addressing it since its formulation; it remains open in the supplied text.