Missing-face extension conjecture for affine stresses

Let $P\subseteq \mathbb R^d$ be a simplicial polytope of dimension $d\geq 2k-1$, with $k\geq 2$. Let $G$ be a missing $(k-1)$-face of $\partial P$, and let $F$ be a $(k-1)$-subset of $G$. An affine $k$-stress on $(\partial P\cup\{G\},p)$ is a stress for the complex obtained by adjoining $G$ and using the natural embedding $p$. Missing-face extension conjecture. There exists such a stress $\lambda$ satisfying

$$\lambda_G>0$$

and

$$\lambda_{F\cup u}\leq 0$$

for every $(k-1)$-face $F\cup u$ of $\partial P$. This conjecture is proposed as a stronger statement implying the stress-sign conjecture, and hence the reconstruction conjectures; it remains open in the stated generality.