Full stress-rank conjecture for generic global $(d,k)$-rigidity

Let $(G,p)$ be a generic framework in $\mathbb{R}^d$. Write $DR_k(G,p)$ for the $(d,k)$-rigidity matrix and $\Omega(\sigma)$ for the stress matrix associated with an equilibrium stress $\sigma$. A framework is globally $(d,k)$-rigid when every $(d,k)$-equivalent framework is congruent to it.

Full stress-rank conjecture. If $(G,p)$ is globally $(d,k)$-rigid and $G$ is not complete, then there exists $\sigma \in \ker DR_k(G,p)^T$ such that

$$\operatorname{rank} \Omega(\sigma)=|V|-d+k-1.$$

The conjecture asserts that the full stress rank is necessary for generic global $(d,k)$-rigidity, in analogy with the corresponding characterization for global rigidity. The paper notes that an equilibrium stress of the required rank exists for the projected framework, but that additionally requiring $\sigma \in \ker DR_k(G,p)^T$ imposes the nontrivial conditions $p_i^T\Omega p_i=0$ for $i\in\{d-k+1,\ldots,d\}$; the conjecture remains open.