The maximum number of paths of even length in a planar graph

For graphs $G$ and $H$, let $N(G,H)$ be the number of unlabeled, not necessarily induced copies of $H$ in $G$, and let $f(n,H)$ be the maximum of $N(G,H)$ over all $n$-vertex planar graphs $G$. Ghosh, Gy\H{o}ri, Martin, Paulos, Salia, Xiao and Zamora conjectured that, for every fixed integer $\ell\ge 2$, $$f(n,P_{2\ell+1}) =4\ell\left(\frac{n}{\ell}\right)^{\ell+1}+O(n^\ell).$$ We prove the conjecture, including the stated error term. Along the way, we also settle the Cox--Martin optimization conjecture.