Subquadratic conjecture for odd cycles with an additional forbidden odd cycle

Let $k<l$ be positive integers, and let $ex(n,F,H)$ denote the maximum number of copies of $F$ in an $n$-vertex graph containing no copy of any graph in $H$.

Subquadratic odd-cycle conjecture. There exists an $\epsilon>0$ such that

$$ex(n,C_{2k+1},{\mathscr C}_{2k}\cup\{C_{2l+1}\})=O(n^{2-\epsilon}).$$

The source proves only a quadratic upper bound for this generalized Turán number and conjectures a power saving below $n^2$. The question concerns how strongly forbidding an additional odd cycle reduces the number of copies of an odd cycle.