Milnor's continuity and vanishing conjecture for simplex volumes

Let $\Omega$ be the space of angle assignments of geometric simplices in hyperbolic or spherical $n$-space, and let $V$ denote the corresponding volume function. The boundary of the angle space of Euclidean simplices is viewed inside the closure $\overline{\Omega}$.

Milnor's continuity and vanishing conjecture. The volume function $V$ admits a continuous extension to $\overline{\Omega}$. Furthermore, the points on $\partial\Omega$ where $V$ vanishes are precisely those which also lie in the closure of the set of angle assignments of Euclidean simplices.

Milnor's conjecture has two parts: the continuity assertion and the characterization of the boundary points where the volume vanishes. The source states that F. Luo had already proved the continuity part and that the paper gives a proof of both parts, so the conjecture is solved.