The chordal graph k-path vertex cover conjecture

Let $k\ge 3$, let $G$ be a chordal graph with clique number $\omega$, and write $n=|V(G)|$. Let $\psi_k(G)$ denote the minimum size of a vertex set meeting every path on $k$ vertices in $G$.

Chordal graph k-path vertex cover conjecture.

$$\psi_k(G) \le \frac{\omega-1}{\omega+k-2}n.$$

The authors present this as a hoped-for improvement to some of the paper's bounds for chordal graphs. No resolution is given in the supplied text.