The vertex-barycentre conjecture for permutahedra and associahedra

Let $W$ be a Coxeter group and let $c\in W$ be a Coxeter element. Choose a real number $a>0$ and set

$$\pmb a=\sum_{s\in S}av_s$$

to fix a realization of the permutahedron ${\mathsf{Perm}}^{\pmb a}(W)$. Let ${\mathsf{Asso}}^{\pmb a}_c(W)$ denote the corresponding $c$-associahedron, and let the vertex barycentre mean the average of the vertices of a polytope.

Vertex-barycentre conjecture. The vertex barycentres of ${\mathsf{Perm}}^{\pmb a}(W)$ and ${\mathsf{Asso}}^{\pmb a}_c(W)$ coincide.

The conjecture is motivated by observations for Loday's realization and other type $A$ and $B$ associahedra. It was supported by computations in the listed finite types, but the source states that these observations had not been proved; no resolution is supplied here.