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 $\text{ex}(n,C_{2\mathrm{\ell }},C_{2k})=\mathrm{\Theta }(n^{\mathrm{\ell }})$ for $k>\mathrm{\ell }\ge 2$. They determined the leading term for $\mathrm{\ell }=2$, but for $k>\mathrm{\ell }\ge 3$ their general lower and upper bounds had different leading constants, leaving open the problem of closing this gap. We solve this problem by showing that, for every $k>\mathrm{\ell }\ge 2$, $$\text{ex}(n,C_{2\mathrm{\ell }},C_{2k})=(\frac{(k-1{)}_{\mathrm{\ell }}}{2\mathrm{\ell }}+o(1))n^{\mathrm{\ell }},$$ where $(k-1{)}_{\mathrm{\ell }}=(k-1)(k-2)\cdots (k-\mathrm{\ell })$.