The finite Garside-shadow conjecture for n-low elements

Let $(W,S)$ be a Coxeter system. For $n\in\mathbb N$, define the dominance depth of a positive root as the number of positive roots it strictly dominates, let the $n$-small roots be those of dominance depth at most $n$, and let $L_n(W)$ be the set of elements whose left inversion sets are the conic hulls of sets of $n$-small roots. A Garside shadow is a subset of $W$ containing $S$ and closed under joins in the right weak order and under taking suffixes. The finite Garside-shadow conjecture for n-low elements. If $n\in\mathbb N$, then $L_n(W)$ is a finite Garside shadow in $(W,S)$. The paper proves that $L_n(W)$ is finite and closed under joins; stability under suffixes is the remaining conjectural part, and the result would yield an infinite filtration by finite Garside shadows.