Sign-vector reconstruction conjecture for simplicial polytopes

Let $P\subseteq \mathbb R^d$ be a simplicial $d$-polytope. For an affine $k$-stress on $P$, record the signs of the coefficients in its squarefree part on the $(k-1)$-faces; let ${\mathcal V}_k(P)$ be the resulting set of sign vectors. Sign-vector reconstruction conjecture. If $k\geq 2$ and $d\geq 2k$, then the $(k-1)$-skeleton of $\partial P$ together with ${\mathcal V}_k(P)$ determines the entire complex $\partial P$. This strengthens Kalai's conjecture by retaining only sign information; it is proved in the paper for $k=2$, while the general case remains open.