The arbitrary-Newton-polygon conjecture for generic specializations

Let $W$ be the Weyl group of the general linear group $GL_h$, and let ${}^J W$ denote the subset parametrizing ${\rm BT}_1$'s of height $h$ and dimension $d$. For a Newton polygon $\xi$, let $w_\xi$ be the element of ${}^J W$ corresponding to $H(\xi)[p]$. A generic specialization $w'$ of $w$ means $w'\subset w$ and $\ell(w')=\ell(w)-1$. Write $\zeta\prec\xi$ when every point of $\zeta$ lies above or on $\xi$, and call this relation saturated when there is no Newton polygon $\eta$ with $\zeta\precneqq\eta\precneqq\xi$.

The arbitrary-Newton-polygon conjecture. For an arbitrary Newton polygon $\xi$, let $w\in{}^J W$ be a generic specialization of $w_\xi$. Then there exists a Newton polygon $\zeta$ such that $\zeta\prec\xi$ is saturated and $w_\zeta\subset w$.

The theorem establishing this assertion when $\xi$ consists of two segments motivates the proposed generalization; proving it would extend the classification of boundary components from the two-segment case to arbitrary central streams.