Cranston et al.'s revised planar Turán conjecture for cycles

Let $C_k$ be the cycle on $k$ vertices, and let $\operatorname{ex}_{\mathcal{P}}(n,C_k)$ denote the maximum number of edges in an $n$-vertex planar graph containing no copy of $C_k$ as a subgraph. Let $d$ be a constant. Cranston et al.'s revised conjecture. There exists a constant $d$ such that for all $k$ and all sufficiently large $n$, we have

$$\operatorname{ex}_{\mathcal{P}}(n,C_k)\le 3n-6-\frac{dn}{k^{\log_2 3}}.$$

This revised conjecture was proposed after the earlier conjecture of Ghosh et al. was disproved for $k\ge 11$ by Cranston, Lidický, Liu and Shantanam, and independently by Lan and Song. The paper presents it as a proposed asymptotic upper bound for planar Turán numbers of cycles.