The density conjecture for extreme points of quasiperfect Vinberg domains

Let $P$ be a quasiperfect Coxeter polytope of negative type, let $W_P$ be its Coxeter group, and let $\Omega_P$ be its Vinberg domain. The extreme points of the closure $\overline{\Omega_P}$ are dense in the boundary $\partial\Omega_P$, except when $W_P$ is affine of type $\widetilde{A}$.

Extreme-point density conjecture. The extreme points of $\overline{\Omega_P}$ are dense in $\partial\Omega_P$ unless $W_P$ is affine of type $\widetilde{A}$.

This is presented as a further condition whose missing implication would yield the reciprocal in the proximal limit set characterization. Its resolution is not supplied in the given text.