OPEN

Let $h(n)$ be minimal such that, for every graph $G$ on $n$ vertices, there is a set of vertices $X$ of size $\lvert X\rvert\leq h(n)$ such that every maximal clique (on at least $2$ vertices) in $G$ contains at least one vertex from $X$.
Let $H(n)$ be maximal such that every triangle-free graph on $n$ vertices contains an independent set on $H(n)$ vertices.
Does $h(n)=n-H(n)$?

It is easy to see that $h(n)\leq n-\sqrt{n}$ and that $h(n)\leq n-H(n)$. Conjectured by Erdős and Gallai, who were unable to make progress even assuming $G$ is $K_4$-free. Erdős remarked that this conjecture is 'perhaps completely wrongheaded'.