More exactly, written in coordinates, I'm curious to know if for every point set $(x_i,y_i)$ there are $(x_i,y_i,z_i,w_i)$ that are vertices of a neighborly polytope.

This problem comes from a simple observation: every planar point set is projection of a convex 3-polytope. One can lift $(x,y)\mapsto (x,y,x^2+y^2)$ as the inverse of projection.

There is the oriented matroid version of such problem, can topes of every acyclic rank-3 uniform oriented matroid realized as a subset of topes of rank-5 neighborly oriented matroid? (~~I'm also not very sure the whether taking subset and projection is always the same~~, this is the strong map conjecture) I'm looking for contour examples for both problem.

neighborlyrequirement. My oversight. $\endgroup$