Higher-dimensional ℓp global rigidity and identifiability conjecture

Let $p$ be an even positive integer with $p\neq 2$, let $n\geq 3$, and let $G$ be a connected graph with $n$ vertices. Write ${\cal CM}_n^p$ for the relevant $\ell_p$-configuration variety and $\pi_G$ for its coordinate projection. Higher-dimensional ℓp conjecture. The following are equivalent:

1. Some/every generic $d$-dimensional framework of $G$ is globally rigid in $d$-dimensional $\ell_p$-space.
2. $\overline{\pi_G({\cal CM}^p_n)}$ is $d$-identifiable.
3. $\overline{\pi_G({\cal CM}^p_n)}$ is $d$-tangentially weakly nondefective.
4. $G$ is 2-connected and $G-e$ contains $d$ edge-disjoint spanning trees for every $e\in E(G)$.
5. $G$ is 2-connected and $G-e$ is locally rigid in the $\ell_p$-plane for every $e\in E(G)$.

The displayed equivalence is proved in the paper for $d=2$, including the generic global-rigidity problem in $\ell_p$-planes. The conjecture proposes its validity in general dimension.