Kalai's affine-stress support conjecture for simplicial spheres

Let $2\leq i\leq d/2$. Let $\Delta$ be either the boundary complex of a simplicial $d$-polytope with its natural embedding $p$, or a simplicial $(d-1)$-sphere, or more generally a normal $(d-1)$-pseudomanifold without boundary, with a generic embedding $p$. Assume that $\Delta$ has no missing faces of dimension at least $d-i+1$.

Kalai's affine-stress support conjecture. Every $(i-1)$-face of $\Delta$ participates in some affine $i$-stress on $(\Delta,p)$. In particular, the space of affine $i$-stresses determines the $(i-1)$-skeleton of $\Delta$.

For generic embeddings, the preceding result for affine $2$-stresses motivates this higher-dimensional support conjecture. Its general case is left open in the source.