Eventual stabilization of saturation numbers for finite cycle intervals

Let $\mathcal{C}_I$ denote the family of cycles whose lengths belong to an integer set $I$, and let $\operatorname{sat}(n,\mathcal{C}_I)$ be the minimum number of edges in an $n$-vertex $\mathcal{C}_I$-saturated graph. The conjecture. There exist functions $r(s)$ on $s\in[4,+\infty)$ and $n(s,r)$ for $s\in[4,+\infty)$ and $r\in[r(s),+\infty)$ such that for any integers $s\ge4$, $r\ge r(s)$ and $n\ge n(r,s)$,

$$\operatorname{sat}(n, \mathcal{C}_{[s,r]})=\operatorname{sat}(n, \mathcal{C}_{[s,+\infty)}).$$

This generalizes the preceding finite-interval conjectures; no resolution is supplied in the source.