Asymptotic normality conjecture for the chromatic number of random graphs

Let $G_{n,p}$ be the binomial random graph with constant $p\in(0,1-1/e^2]$, and let $Y_n=\chi(G_{n,p})$. Call $n$ good when it is outside the paper's designated bad range, namely when $\mu_{\alpha(n)}(n)$ is not between $n^{1/2-\delta}$ and $n^{1/2+\delta}$ for the fixed constant $\delta>0$. Let $\lambda(n)$ be the exponent from the Zigzag Conjecture.

Asymptotic normality conjecture. There are functions $f(n)$ and $g(n)$ such that, at least for good $n$,

$$\frac{Y_n-f(n)}{g(n)}\overset{\mathrm d}{\longrightarrow}N(0,1),$$

where $N(0,1)$ is standard Gaussian, and

$$g(n)=n^{\lambda(n)+o(1)}.$$

The conjecture refines the proposed fluctuation scale by asserting a Gaussian limit, but leaves the transition points called bad unresolved.