The induced $K_4$-minor-free subgraph conjecture for planar graphs

Let $G$ be a planar graph of order $n$. The $K_4$-minor-free subgraph number $s_{K_4}(G)$ is the maximum order of an induced $K_4$-minor-free subgraph of $G$.

Induced $K_4$-minor-free subgraph conjecture.

$$s_{K_4}(G)\ge\frac{2n}{3}.$$

This conjecture is motivated by the corresponding conjecture for induced outerplanar subgraphs and remains open.