Two-consecutive-values conjecture for the chromatic number of sparse random Borsuk graphs

Let $G(n,\alpha)$ be the random Borsuk graph in dimension $d$, let $\chi(G(n,\alpha))$ denote its chromatic number, and let $\alpha_0$ be the constant supplied by the Erdős–Hajnal result mentioned in the paper's introduction. Two-consecutive-values conjecture. For all $d\geq 1$ and all sequences $\alpha=\alpha(n)$ with $\alpha<\alpha_0$, there is a sequence of integers $k=k(n)$ such that

$$\mathbb{P}\bigl(\chi(G(n,\alpha))\in\{k,k+1\}\bigr)=1-o_n(1).$$

This is presented as a weaker conjecture that might be approachable without first proving the hitting-angle conjecture. It predicts concentration of the chromatic number on two consecutive values in the indicated sparse regime and remains open.