Füredi–Kim conjecture on saturation numbers of long cycles

Let $C_k$ denote the cycle graph on $k$ vertices, and let $\mathrm{sat}(n,C_k)$ be its saturation number. Füredi–Kim conjecture. There exists a constant $k_0$ such that

$$\mathrm{sat}(n,C_k)=\frac{k-3}{k-4}n+O(k^2)$$

holds for all integers $k\geqslant k_0$. The conjecture is partially resolved in the source: for each fixed even integer $k\geqslant28$, the asserted asymptotic holds with $O(1)$ in place of $O(k^2)$; the general statement remains open.