Saturation number for cycles of lengths congruent to 1 modulo 3

For an 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. Write $3\mathbb{Z}_++1=\{3k+1:k\in\mathbb{Z}_+\}$. The conjecture. For every integer $n\ge1$,

$$\operatorname{sat}(n, \mathcal{C}_{3\mathbb{Z}_++1})=\left\lceil\frac{5n}{4}-\frac{3}{2}\right\rceil.$$

The paper proves an analogous exact formula for $\mathcal{C}_{2\mathbb{Z}_++2}$ but explicitly presents this assertion as an unresolved conjecture.