Dyer–Hohlweg bijection conjecture for low elements and small inversion sets

Let $(W,S)$ be a Coxeter system with root system $cPhi$, positive roots $cPhi^+$, and inversion set $N(w)$ for $w eqthinspacecin W$. A positive root $a eqthinspacecincPhi^+$ is small if, for every $b eqthinspacecincPhi^+\setminus\{a\}$, there \exists $w eqthinspacecin W$ such that $a eqthinspacecin N(w)$ and $b otin N(w)$. Let $cSigma$ be the set of small roots, let $cSigma(w)=cSigma\cap N(w)$, and let $cLambda$ be the set of all small inversion sets. An element $w eqthinspacecin W$ is low if

$$N(w)=coperatorname{cone}(cSigma(w))\capcPhi^+.$$

Let $L$ be the set of low elements. Dyer–Hohlweg's conjecture. The map

$$w\longmapsto cSigma(w):L\longrightarrowcLambda$$

is a bijection between low elements and small inversion sets. This conjecture asks whether every small inversion set is realized by exactly one low element. The map is already known to be injective because inversion sets determine elements of the Coxeter group; the conjecture concerns surjectivity and its connection with the geometry of Shi regions.