Conjecture on extremal graphs for linearly long induced cycles

Let $c\in(0,1)$, let $l(n)=\lceil cn\rceil$, and let $c(n,l)$ be the maximum number of induced cycles of length $l$ in a graph on $n$ vertices. Let $C(n,l)$ be the set of graphs attaining this maximum, and call a graph a cyclic braid of length $l$ as in the paper. The linear-length induced-cycle conjecture. If $l(n)=\lceil cn\rceil$, then for sufficiently large $n$ the only graphs in $C(n,l)$ are cyclic braids of length $l$. The paper determines the extremal graphs for several related induced-cycle problems, but this linear-length question is presented as an expected statement and remains unresolved.