Eventual saturation number for cycles of lengths from 4 to r

For a positive integer set $I$, let $\mathcal{C}_I$ denote the family of cycles whose lengths belong to $I$, 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. For any integer $r a 5$, there \exists a number $n(r)$ such that for any integer $n a n(r)$,

$$\operatorname{sat}(n, \mathcal{C}_{[4,r]})=\left\lceil\frac{5n}{4}-\frac{3}{2}\right\rceil.$$

The paper establishes the corresponding formula for $I=\{4,5\}$ and for $I=[4,+\infty)$, but the assertion for every finite upper endpoint $r$ remains open.