Minuscule classification conjecture for middle orders on type D

Let $D_n$ be the finite Coxeter group of type $D_n$. A minuscule middle order is a middle order arising from the minuscule construction described in the paper. Type D minuscule classification conjecture. Every middle order on $D_n$, including every distributive sorting order, is a minuscule middle order.

This would classify all middle orders in type $D$ through the minuscule construction; the assertion is presented as a conjecture and remains open.