Odd-cycle saturation conjecture

Let $\mathcal{C}_{2k+1}$ denote the family consisting of the cycle $C_{2k+1}$, and let $\mathrm{sat}(\mathcal{C}_{2k+1};n)$ be its saturation number on $n$ vertices. Odd-cycle saturation conjecture. For all $k\ge 1$ there exists a $c_k>0$ such that

$$\mathrm{sat}(\mathcal{C}_{2k+1};n)\le \left(\frac{1}{4}-c_k\right)n^2+o(n^2).$$

This conjecture predicts a strict improvement over the general quadratic upper bound for saturation by every odd cycle. The source presents it as an expected stronger bound; no resolution is given.