Extremal conjecture for equal-degree endpoint paths of odd length

Let $p_{\ell}(N)$ denote the maximum number of edges in a graph on $N$ vertices containing no path of length $\ell$ whose endpoints have equal degree. For odd integers $\ell\geq 3$ and sufficiently large positive integers $n$, consider graphs on $2n+1$ vertices; the complete bipartite graph $K_{n,n+1}$ has $n^2+n$ edges.

Odd-length extremal conjecture. For any odd integer $\ell\geq 3$ and sufficiently large $n$, it holds that

$$p_{\ell}(2n+1)=n^2+n.$$

Equivalently, $K_{n,n+1}$ remains extremal for paths of every odd length at least three. This extends the problem of Erdős and Hajnal on paths with equal-degree endpoints; the asserted extremality is posed as an open problem for general odd lengths.