Asymptotic planar Turán conjecture for cycles of length at least seven

Let $G$ be an $n$-vertex $C_{\ell}$-free plane graph, meaning that $G$ contains no cycle of length $\ell$, where $\ell\geq 7$. Write $e(G)$ for the number of edges of $G$. Asymptotic planar Turán conjecture. There exists an integer $N_0>0$ such that, whenever $n\geq N_0$,

$$e(G)\leq \frac{3(\ell-1)}{\ell}n-\frac{6(\ell+1)}{\ell}.$$

This conjecture gives the proposed asymptotic upper bound for the number of edges in an $n$-vertex plane graph avoiding a cycle of length $\ell$. The supplied source states the claim for every $\ell\geq 7$, but provides no resolution status here.