Cranston–Lidický–Liu–Shantanam conjecture on planar Turán numbers of cycles

Let $C_k$ denote the cycle of length $k$, and let $ex_{\mathcal{P}}(n,C_k)$ be the maximum number of edges in an $n$-vertex planar graph containing no copy of $C_k$. Cranston–Lidický–Liu–Shantanam conjecture. There exists a constant $D$ such that, for all $k$ and all sufficiently large $n$,

$$ex_{\mathcal{P}}(n,C_k)\le \left(3-\frac{3}{Dk^{\lg_2^3}}\right)n.$$

This conjecture is proposed after the preceding conjecture was disproved for every $k\ge 11$. The supplied text gives no resolution of this revised asymptotic upper bound.