Logarithmic threshold conjecture for the chromatic number 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. Logarithmic threshold conjecture. For every $d\geq 2$ there exists a constant $c=c(d)$ such that, for every fixed $\varepsilon>0$ and every sequence $\alpha=\alpha(n)$,

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

The paper identifies the logarithmic regime as the relevant one for chromatic number at least $d+2$ and suggests a connection with coverage of the sphere by random caps. Establishing the claimed threshold and its constant remains open.