The proximal limit set characterization for Coxeter polytopes of negative type

Let $P$ be a Coxeter polytope of negative type. Write $\Omega_P$ for its Vinberg domain and $\Lambda_P$ for the proximal limit set of its reflection group. The polytope $P$ is large quasiperfect when it has the corresponding large quasiperfectness property.

Proximal limit set characterization.

$$\partial\Omega_P=\Lambda_P$$

if and only if $P$ is large quasiperfect.

The forward implication is established in the preceding corollary, while the conjectured reciprocal would characterize exactly when the proximal limit set fills the boundary of the Vinberg domain for these reflection groups.