Saturation number for cycles of lengths congruent to 2 modulo a

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 $a\mathbb{Z}_++2=\{ak+2:k\in\mathbb{Z}_+\}$. The conjecture. For integers $a\ge2$ and $n\ge a+1$,

$$\operatorname{sat}(n, \mathcal{C}_{a\mathbb{Z}_++2})=n+\binom{a}{2}-1.$$

The preceding fact supplies the matching upper bound, but the source does not state a proof of the required lower bound; this exact formula is therefore left as a conjecture.