A simple generalized permutahedron conjecture for Speer fans

Let $E$ be a finite set. Let $\overline{GP}_{E}$ be the set of generalized permutahedra on $E$ up to normal equivalence, and let $\overline{SGP}_{E}$ be the subset consisting of polytopes whose connected components are simple. Write $\Sigma_P(1)$ for the relevant fan associated with a generalized permutahedron $P$.

Simple generalized permutahedron conjecture. There is a natural map

$$\overline{GP}_{E}\longrightarrow \overline{SGP}_{E},\qquad P\longmapsto P^{s},$$

which commutes with contraction and restriction and satisfies

$$\Sigma_{P}(1)=\Sigma_{P^{s}}(1).$$

The conjecture seeks a generalization of Speer's construction that applies to all generalized permutahedra while retaining the associated fan. The source does not provide evidence that this conjecture has been resolved.