Worst-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

$$w_n=\frac{\sqrt n\log\log n}{\log^3 n}.$$

Worst-case variance conjecture. One has

$$0<\limsup\frac{\sigma_n}{w_n}<\infty.$$

Moreover, for every constant $c>0$ there is a constant $d>0$ such that, along every sequence of integers $n$ satisfying $\mu_{\alpha(n)}(n)\sim c n/\log^2 n$, one has $\sigma_n\sim d w_n$. This conjecture proposes that the polylogarithmic lower bound for the variance has the correct order in the worst case; the paper notes that one inequality is already implied by a theorem subject to an announced result.