Kalai's affine-stress reconstruction conjecture up to affine type

Let $d\geq 4$, let $2\leq i\leq d/2$, and let $P$ be a simplicial $d$-polytope. A missing face is a vertex set whose proper subsets are faces but which is not itself a face.

Kalai's affine-type reconstruction conjecture. If $P$ has no missing $(d-1)$-faces, then its graph and the space of affine $2$-stresses determine $P$ up to affine equivalence. More generally, if $P$ has no missing faces of dimension at least $d-i+1$, then the space of affine $i$-stresses determines $P$ up to affine equivalence.

This strengthens reconstruction of combinatorial type by seeking the affine realization itself. The source gives no resolution.