Characterization of spherical orbit spaces from affine colorings

Let $P$ be a simple $n$-polytope, let $\boldsymbol\lambda$ be an affine coloring of $P$ of rank $r$, and let $\mathcal{C}(P,\lambda)$ and $\mathcal{C}(n,r+1)$ denote the associated complexes. Let $N(P,\lambda)$ be the space corresponding to this affine coloring.

Spherical orbit-space conjecture. The space $N(P,\lambda)$ is homeomorphic to $S^n$ if and only if

$$\mathcal{C}(P,\lambda)\simeq\mathcal{C}(n,r+1).$$

The claim gives a combinatorial characterization of when the orbit space associated with an affine coloring is an $n$-sphere. The supplied text does not indicate whether this statement has been proved or disproved.