Sturmfels–Ziegler extension-space conjecture for realizable oriented matroids

Let $\mathcal{M}$ be a realizable oriented matroid of rank $r$. Its extension space is the simplicial complex associated to the natural poset of one-point extensions of $\mathcal{M}$. Sturmfels–Ziegler extension-space conjecture. The extension space of $\mathcal{M}$ is homotopy equivalent to a sphere of dimension $r-1$. The source presents this as an open conjecture relevant to the study of lifting triangulations and oriented matroids; no resolution is given in the supplied text.