Affine stress support conjecture for spheres without large missing faces

Let $2\leq i\leq d/2$, let $\Delta\in S(d-i,d-1)$, and let $p$ be a generic or natural embedding of $\Delta$ in $\mathbb{R}^d$. An affine $i$-stress is an element of the affine stress space of $(\Delta,p)$. Affine stress support conjecture. Every $(i-1)$-face of $\Delta$ participates in some affine $i$-stress on $(\Delta,p)$.

The paper states that this conjecture is disproved, so it is false in the stated generality.