Akiyama–Watanabe conjecture on induced forests in bipartite planar graphs

Let $G$ be a simple bipartite planar graph. Write $|G|=|V(G)|$, and let $a(G)$ denote the maximum number of vertices in an induced forest of $G$. Akiyama–Watanabe conjecture.

$$a(G) \geq \frac{5|G|}{8}.$$

The conjecture was proposed in 1987. The paper proves the weaker lower bound $a(G)\geq \lceil(4|G|+3)/7\rceil$, so the stated $5|G|/8$ bound is not resolved by the supplied text.