The density conjecture for circular chromatic numbers of random graphs

Let $G_{n,\frac{c}{n}}$ be the random graph with edge probability $\frac{c}{n}$, and let $\chi_{\mathrm{c}}(G)$ denote its circular chromatic number. Density conjecture. There are no real numbers $2\leq a<b$ such that, for every value of $c$, the probability

$$\operatorname{Pr}\left(\chi_{\mathrm{c}}\left(G_{n,\frac{c}{n}}\right)\in(a,b)\right)\to 0.$$

This asserts that the circular chromatic numbers of sparse random graphs do not omit any fixed interval in $[2,\infty)$ in the limiting probabilistic sense; it is presented as the conjecture that these values should be dense.