The equality conjecture for Colin de Verdière-type parameters and realizable dimension

Let $G$ be a multigraph. The parameters $\lambda(G)$ and $\operatorname{rd}(G)$ denote, respectively, the parameter introduced earlier in the paper and the realizable dimension of $G$.

Equality conjecture.

$$\lambda(G)=\operatorname{rd}(G).$$

No example is currently known that separates these parameters. If the equality holds, it has an important implication for the maximum dimension of globally rigid, super stable, and universally rigid tensegrity realizations.