The surjectivity conjecture for generalized W-permutohedra

Let $W$ be the Weyl group and $c$ a Coxeter element. The cone of generalized $W$-permutohedra projects to the cone of $(W,c)$-prepolypositroids, and the $(W,c)$-twisted alcoved envelope gives the corresponding map described in Theorem.

Surjectivity conjecture for generalized W-permutohedra. The maps in Theorem are surjective.

This conjecture concerns whether the constructions relating generalized $W$-permutohedra and $(W,c)$-prepolypositroids attain every object in their target cones. The supplied text gives no resolution status.