Novik–Zheng higher affine-stress reconstruction conjecture

Let $d\geq 4$, let $2\leq i\leq d/2$, and let $P$ be a simplicial $d$-polytope. Assume that $P$ has no missing faces of dimension at least $d-i+1$. Equivalently, let $R={\mathbb R}[x_1,\dots,x_n]$, let $I_P$ be the Stanley–Reisner ideal of the boundary complex of $P$, let $\Theta_P=\theta_1,\dots,\theta_d$ be the linear forms determined by the vertex coordinates, and let $\ell=x_1+\cdots+x_n$; then the alternative hypothesis is that $I_P$ has no generators of degree at least $d-i+2$. The higher affine-stress reconstruction conjecture. The affine type of $P$ can be reconstructed from the space of affine $i$-stresses of $P$. Equivalently, under the algebraic hypothesis, $(\Theta_P,\ell)_1$ is determined by $\big(I_P+(\Theta_P,\ell)\big)_i$. This generalizes the affine $2$-stress reconstruction problem. The case $i=2$ is the original conjecture above, while the general cases remain open; in particular, the source notes that the case $d=2i$ remains open.