The Sigma-map bijectivity conjecture for n-low elements

Let $(W,S)$ be a Coxeter system. Let $L_n(W)$ denote the set of $n$-low elements and let $\Lambda_n(W)$ denote the corresponding set of admissible collections of $n$-small roots; for each $w\in L_n(W)$, let $\Sigma_n(w)$ be the associated collection of roots. The Sigma-map bijectivity conjecture. The map

$$\Sigma_n:L_n(W)\to\Lambda_n(W)$$

is a bijection. This is motivated by examples in which the map is bijective; the general status is not resolved in the supplied text.