Kalai's conjecture on recovering lower affine-stress spaces

Let $1\leq j<i\leq d/2$, and let $P$ be a simplicial $d$-polytope whose boundary complex has no missing faces of dimension at least $d-i+1$. Write $\mathcal S^a_k(P)$ for the space of affine $k$-stresses of $P$.

Kalai's lower-stress reconstruction conjecture. The space $\mathcal S^a_i(P)$ determines the space $\mathcal S^a_j(P)$.

The conjecture proposes that higher affine stresses contain enough information to recover all lower stress spaces under a no-large-missing-faces hypothesis. The source gives no resolution.