The affine partition-of-unity conjecture for simplicial polytopes and homology spheres

Let $2\leq i\leq (d-1)/2$. Let $(\Delta,p)$ be either the boundary complex of a simplicial $d$-polytope with its natural embedding $p$, or a $\mathbb Z/2\mathbb Z$-homology $(d-1)$-sphere with a generic embedding $p$. Write $\mathcal S^a_i(\Delta,p)$ for the space of affine $i$-stresses and $\operatorname{st}(v)$ for the star of $v$.

Affine partition-of-unity conjecture.

$$\mathcal S^a_i(\Delta,p)=\sum_{v\in V(\Delta)}\mathcal S^a_i(\operatorname{st}(v),p).$$

This asks whether every affine $i$-stress is generated by stresses supported on vertex stars, paralleling the known linear-stress partition-of-unity result. The source gives no resolution.