The planar 3-path vertex cover conjecture

Let $G$ be a planar graph of order $n$, and let $\psi_3(G)$ denote its 3-path vertex cover number.

Planar 3-path vertex cover conjecture.

$$\psi_3(G) \le \frac{2}{3}n.$$

This conjecture was first mentioned by Brešar et al. in 2011. Equality is attained by, for example, vertex-disjoint unions of octahedron graphs arranged in a planar way; the conjecture remains open in the source.