Stress-sign conjecture for missing faces of simplicial polytopes

Let $P$ be a simplicial $d$-polytope, and let $M$ be a missing face of $\partial P$ satisfying

$$k\geq 2,\qquad d\geq 2k,\qquad k\leq \dim M\leq d-k.$$

Stress-sign conjecture. There exists a $(k-2)$-face $F\subset M$ and an affine $k$-stress $\lambda$ on $P$ such that, for every $(k-1)$-face $G=F\cup v_{F,G}$ of $\partial P$, one has $\lambda_G>0$ if $v_{F,G}\in M$, whereas $\lambda_G\leq 0$ if $v_{F,G}\notin M$. If true, this gives sign-stress certificates for missing faces and implies the sign-vector reconstruction conjecture; the paper proves the latter implication but leaves this conjecture open.