Erdős Problem #1156 — Non-concentration of the random-graph chromatic number

For $G\sim G(n,1/2)$, can $\chi(G)$ be concentrated with high probability on a bounded number of values? More strongly, is there a function $\omega(n)\to\infty$ such that for every deterministic $f(n)$ one has

$$\mathbb{P}(|\chi(G)-f(n)|<\omega(n))<1/2$$

for all sufficiently large $n$?