Best-case variance conjecture for the chromatic number of random graphs

Let $G_{n,p}$ be the binomial random graph with constant $p\in(0,1-1/e^2]$, let $Y_n=\chi(G_{n,p})$, and let $\sigma_n^2=\operatorname{Var}(Y_n)$. Define

$$\widetilde w_n=\frac{n^{1/4}}{\log^{7/4}n}.$$

Best-case variance conjecture. One has

$$0<\liminf\frac{\sigma_n}{\widetilde w_n}<\infty.$$

This is the proposed lower-order scale for the variance in the best case. The authors explicitly describe this conjecture as less certain than the worst-case statement and give no resolution.