The circumference conjecture for essentially 4-connected planar graphs

Let $G$ be an essentially $4$-connected planar graph on $n$ vertices, and let $\operatorname{circ}(G)$ denote the length of a longest cycle in $G$. Circumference conjecture. Every essentially $4$-connected planar graph on $n$ vertices satisfies

$$\operatorname{circ}(G)\ge \frac{2}{3}(n+4).$$

The bound is motivated by constructions of essentially $4$-connected maximal planar graphs attaining equality. The source states that it remains open whether an essentially $4$-connected planar graph can have smaller circumference; the paper proves the bound for essentially $4$-connected maximal planar graphs.