The general-dimensional identifiability characterization of global rigidity

Let $p$ be an even positive integer with $p\neq 2$, let $d$ be a positive integer, and let $G$ be a connected graph with $n\geq 3$ vertices. Let $\overline{\pi_G(\mathcal{CM}^p_n)}$ denote the relevant projected $p$-Cayley–Menger variety, and let $d$-identifiability and $d$-tangential weak defectivity have their usual meanings.

General-dimensional characterization conjecture. The following conditions are equivalent:

1. A/every generic $d$-dimensional framework $(G,\bm p)$ is globally rigid in $\ell_p^d$.
2. $\overline{\pi_G(\mathcal{CM}^p_n)}$ is $d$-identifiable.
3. $\overline{\pi_G(\mathcal{CM}^p_n)}$ is not $d$-tangentially weakly defective.
4. A generic $d$-dimensional framework $(G,\bm p)$ has a $k$-th coordinated self-stress $\omega^k$ such that $\operatorname{rank}L_{G,\omega^k}=n-2$ for all $k$.
5. $G$ is 2-connected and redundantly $d$-tree-connected.

The equivalence is proved in the paper for the plane, and extending the result to general dimension is explicitly identified as an open problem.