Reiner–Tenner–Yong's CDE conjecture for vexillary permutations

Let $\lambda$ be a rectangular staircase of the form

$$\lambda=\delta_d\circ b^a.$$

Let $w\in\mathfrak{S}_n$ be a vexillary permutation of shape $\lambda$, meaning that its Rothe diagram can be transformed to $\lambda$ by a permutation of rows and columns. Reiner–Tenner–Yong's CDE conjecture. The initial weak order interval $[e,w]$ is CDE with edge density

$$\frac{(d-1)ab}{a+b}.$$

This conjecture generalizes the known result for dominant permutations, which are the essentially unique vexillary permutations whose Rothe diagrams equal the prescribed shape. It extends the family of known CDE posets beyond distributive lattices and the previously established dominant weak-order intervals.