The optimal-size clique transversal conjecture for 4-chordal graphs

Let $G$ be a chordal graph with $n$ vertices. Call $G$ 4-chordal if every edge of $G$ is contained in a clique with four vertices, and call a set of vertices $U$ a clique transversal if it meets every non-trivial maximal clique of $G$.

Optimal-size clique transversal conjecture. Every $4$-chordal graph with $n$ vertices has a clique transversal of size at most $2n/7$.

The question generalizes known bounds for $2$- and $3$-chordal graphs. Andreae and Flotow constructed arbitrarily large examples requiring at least $2n/7-O(1)$ vertices, suggesting that the bound is asymptotically tight; the paper proves the stronger integral bound $2(n-1)/7$ for $n\geq 5$.