Stable line shelling conjecture for cubical polytopes

Let $\mathcal{C}$ be the boundary complex of a cubical polytope. A stable line shelling is a line shelling satisfying the stability condition defined through the realization and induced hyperplane arrangements associated with $\mathcal{C}$. Stable line shelling conjecture. The boundary complex of any cubical polytope admits a stable line shelling. This conjecture proposes that every cubical polytope has a shelling with the required geometric stability, strengthening the existence of a shelling by imposing compatibility with a suitable realization and line. The supplied text gives this as a hope and does not state any resolution, so its status remains open.