Proper elements of Coxeter Groups

We extend the notion of proper elements to all Coxeter groups. For all infinite families of Coxeter groups we prove that the probability a random element is proper goes to zero in the limit. This proves a conjecture of the third author and A. Yong regarding the proportion of Schubert varieties that are Levi spherical for all infinite families of Weyl groups. In the exceptional families we enumerate the proper elements.