Conditional non-degeneracy conjecture for spherical simplices

Fix $k\geq 2$, and let $\mathcal W_{d+k,d}$ be the random polyhedral cone in the paper's model. Write $\alpha(\mathcal W_{d+k,d})$ for its solid angle. Condition on the event that

$$\mathcal W_{d+k,d}\cap\mathbb S^{d-1}$$

is a spherical simplex.

Conditional non-degeneracy conjecture. For fixed $k\geq 2$, the conditional distribution of $\alpha(\mathcal W_{d+k,d})$ converges weakly as $d\to\infty$ to a limit law not concentrated at $0$.

The conjecture predicts that the rare spherical-simplex event is caused by a cone with solid angle of constant order, rather than by angles collapsing to zero. No resolution is stated in the supplied text.