A simplicial $d$-sphere is a simplicial complex homeomorphic to the $d$-sphere. It is known that not every such complex can be embedded into $\Bbb R^{d+1}$ as the boundary complex of a convex simplicial polytope. But what if we drop convexity (but keep that the embedded simplices must be flat)?
The paper Nonconvex Embeddings of the Exceptional Simplicial 3-Spheres with 8 Vertices by Mihalisin and Williams states this as a conjecture:
Conjecture 5.2. Every simplicial $d$-sphere is linearly embeddable in $\Bbb R^{d+1}$.
Has there been any progress on this question since 2002 when this paper appeared? For example, has this been verified for simplicial 3-spheres on 9 vertices?
