Zhu et al.'s generalized Turán conjecture for cycles with bounded circumference

For a graph $G$, let $N(C_q,G)$ denote the number of copies of the cycle $C_q$ in $G$, and let $\operatorname{ex}(n,C_q,\mathcal C_{\ge L+1})$ be the maximum possible value of $N(C_q,G)$ over all $n$-vertex graphs $G$ containing no cycle of length at least $L+1$. Let $H(n,L)$ be the Kopylov-type extremal graph on $n$ vertices with circumference at most $L$. Zhu et al.'s conjecture. For fixed integers $q\ge4$ and $L>q$, and all sufficiently large $n$,

$$\operatorname{ex}\bigl(n,C_q,\mathcal C_{\ge L+1}\bigr) =N\bigl(C_q,H(n,L)\bigr).$$

This conjecture proposes that the Kopylov-type construction is asymptotically the exact extremal graph for counting copies of $C_q$ under a bounded-circumference restriction. The source indicates that the result is known for $q=4$ and $q=5$, while the stated general case remains open.