Ghosh et al.'s conjecture on even-length paths in planar graphs

For a finite simple graph $G$, let $N(G,H)$ be the number of unlabeled, not necessarily induced copies of $H$ in $G$. Define

$$f(n,H)=\max\{N(G,H): |V(G)|=n \text{ and } G \text{ is planar}\}.$$

Let $P_k$ denote the path on $k$ vertices, so $P_{2\ell+1}$ has even length $2\ell$. Ghosh et al.'s conjecture. For every fixed integer $\ell\ge 2$,

$$f(n,P_{2\ell+1})=4\ell\left(\frac{n}{\ell}\right)^{\ell+1}+O(n^\ell).$$

The lower bound is supplied by blowing up one bipartition class of a cycle $C_{2\ell}$, while the conjectured matching upper bound remained to be proved in the source. The corresponding conjecture has since been resolved by Lv, Győri, He, Salia, Tompkins and Zhu.