Polynomial threshold conjecture for intermediate chromatic numbers of random Borsuk graphs

Let $G(n,\alpha)$ be the random Borsuk graph in dimension $d$, and let $\chi(G(n,\alpha))$ denote its chromatic number. Polynomial threshold conjecture. For each $d\geq 2$, there exist constants $c_3=c_3(d),\dots,c_d=c_d(d)$ such that

$$c_2<c_3<\dots<c_d,$$

where $c_2$ is the constant supplied by the paper's theorem, and, for every fixed $\varepsilon>0$, every $3\leq k\leq d$, and every sequence $\alpha=\alpha(n)$,

$$\lim_{n\to\infty}\mathbb{P}\bigl(\chi(G(n,\alpha))>k\bigr)= \begin{cases} 1 & \text{if }\alpha>(1+\varepsilon)c_k n^{-1/d},\\ 0 & \text{if }\alpha<(1-\varepsilon)c_k n^{-1/d}. \end{cases}$$

The claim gives the expected sharp thresholds for the intermediate chromatic-number transitions, refining the paper's almost-all-$n$ result. The constants and the asserted sharpness are not proved.