Asymptotic conjecture for the number of even-length paths in planar graphs

Let $P_k$ be a path of length $k$, and let $f(n,P_k)$ denote the maximum number of copies of $P_k$ in a planar graph on $n$ vertices. Even-length path conjecture. For every positive integer $\ell$,

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

This conjectures the asymptotic value of the generalized extremal function for even-length paths; the source proposes it together with the corresponding odd-length formula, and no resolution is supplied here.