Hitting-angle conjecture for chromatic number and neighbourhood complexes

Fix $d\geq 2$ and $k\in\{3,\dots,d+1\}$. In the random Borsuk graph process, edges are added in order of decreasing geodesic distance as the angle parameter $\alpha$ increases. For a monotone graph property $\mathcal{R}$, let $\alpha(n,\mathcal{R})$ be the random angle at which the process first attains $\mathcal{R}$. Let $\mathcal{P}$ be the property of having chromatic number $>k$, and let $\mathcal{Q}$ be the property of containing a subgraph whose neighbourhood complex is homotopy equivalent to $S^{k-1}$. Hitting-angle conjecture.

$$\mathbb{P}\bigl(\alpha(n,\mathcal{P})=\alpha(n,\mathcal{Q})\bigr)=1-o_n(1).$$

The conjecture proposes that the chromatic transition is asymptotically witnessed exactly when the relevant topological subgraph first appears. The paper notes that the analogous assertion for $k=2$ holds, but the stated range remains open.