Odd-cycle generalized Turán conjecture for forbidden smaller odd cycles

For integers $k$ and $\ell$, write $\operatorname{ex}(n,C_k,C_\ell)$ for the maximum number of copies of the cycle $C_k$ in an $n$-vertex graph containing no copy of $C_\ell$. Let $k>\ell\geq 3$ be odd integers. Odd-cycle generalized Turán conjecture. It holds that

$$\operatorname{ex}(n,C_{k}, C_{\ell}) = \left(\binom{k}{\frac{k-(\ell+2)}{2}} + \binom{k}{\frac{k-3(\ell+2)}{2}} + \binom{k}{\frac{k-5(\ell+2)}{2}} + \ldots\right)\frac{n^k}{(\ell+2)^k} + o(n^k).$$

The conjectured extremal construction is a balanced blow-up of an $(\ell+2)$-cycle. The paper proves the corresponding asymptotic result when $k=\ell+2$, but leaves the general odd case as a conjectural extension.