Type B product conjecture for middle orders

Let $B_n$ be the type $B$ finite Coxeter group, with $n\geq3$, and let $J$ be a maximal parabolic subgroup. A middle order is a distributive lattice that refines the weak order and is coarsened by the Bruhat order. Type B product conjecture. Every middle order on $B_n$ can be obtained as the product of a middle order on $W_J$ and a distributive sorting order on ${}^JW$.

This would give a recursive description of all middle orders in type $B$ from maximal parabolic data; it is stated conjecturally and remains open.