Upper-root description of the Cambrian cluster–noncrossing bijection

Let $W$ be a finite Coxeter group, let $c$ be a Coxeter element, and let a $c$-cluster $C$ have upper roots among its roots. Associate to $C$ the intersection of the hyperplanes orthogonal to those upper roots. Upper-root bijection conjecture. The resulting map from $c$-clusters to $c$-noncrossing subspaces is a bijection and agrees with the established map

$$\operatorname{NC}_c\circ\operatorname{cl}_c^{-1}.$$

This conjecture is presented as an immediate consequence of the span-intersection conjecture together with the preceding theorem identifying $\operatorname{NC}_c\circ\operatorname{cl}_c^{-1}$ using the lower roots; the source reports computational verification of the needed orthogonality in rank at most $7$.