Generalized Turán problems for shorter even cycles

For graphs $H$ and $F$, let $\text{ex}(n,H,F)$ denote the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. Gerbner, Győri, Methuku, and Vizer proved that $\text{ex}(n,C_{2\ell},C_{2k})=Θ(n^\ell)$ for $k>\ell\ge2$. They determined the leading term for $\ell=2$, but for $k>\ell\ge3$ th