Lattice conjecture for the c-noncrossing Bruhat order

Let $W$ be a finite Coxeter group and let $c$ be a Coxeter element. The $c$-noncrossing Bruhat order is the order on $W$ defined from the noncrossing Bruhat construction. Bruhat-order lattice conjecture. The $c$-noncrossing Bruhat order on $W$ is a lattice. Together with the proposed isomorphism to the c-noncrossing weak order, this would be equivalent to the corresponding weak-order lattice conjecture; the supplied text gives no general proof.