Isomorphism conjecture for the c-noncrossing weak and Bruhat orders

Let $W$ be a finite Coxeter group and let $c$ be a Coxeter element. Consider the $c$-noncrossing weak order and the $c$-noncrossing Bruhat order on $W$. Order-isomorphism conjecture. The $c$-noncrossing weak order on $W$ and the $c$-noncrossing Bruhat order on $W$ are isomorphic. If true, this would identify the two natural partial orders and would make their lattice conjectures equivalent; the supplied text gives no general proof.