The global rigidity and coordinated self-stress conjecture in $\ell_p^d$

Let $p$ be an even positive integer with $p\neq 2$, let $d$ be a positive integer, and let $(G,\bm p)$ be a generic framework in $\ell_p^d$ with $n\geq 3$ vertices. A $k$-th coordinated self-stress is a self-stress obtained by multiplying the edge weights of a self-stress by the $(p-2)$nd power of the difference of the $k$th coordinates; $L_{G,\omega^k}$ denotes its weighted graph Laplacian.

Global rigidity and self-stress conjecture. The framework $(G,\bm p)$ is globally rigid in $\ell_p^d$ if and only if it has a $k$-th coordinated self-stress $\omega^k$ for some $k$ such that

$$\operatorname{rank}L_{G,\omega^k}=n-2.$$

The paper proves this characterization in the two-dimensional case, while the asserted general-dimensional extension is presented as an open problem.