Planar induced outerplane subgraph conjecture

Let $G$ be a planar graph on $n$ vertices, and let an induced outerplane graph be an induced subgraph whose embedding inherited from $G$ is an outerplanar embedding. Planar outerplane-subgraph conjecture. The graph $G$ contains an induced outerplane graph of size at least

$$\frac{2n}{3}.$$

The source notes that this conjecture was also mentioned by Angelini, Evans, Frati, and Gudmundsson.