Flexible 2-periodic placement characterization for connected gain graphs

Let $G$ be a connected $\mathbb{Z}^2$-gain graph. A full placement-lattice is a pair $(p,L)$ with $p$ a placement of $G$ and $L$ a full lattice in $\mathbb{R}^2$; the associated framework is flexible when it has a nontrivial continuous flex. A type 1 flexible 2-lattice NBAC-colouring, type 2 flexible 2-lattice NBAC-colouring, type 3 flexible 2-lattice NBAC-colouring, and fixed lattice NBAC-colouring are the colouring types specified in the paper, and $\operatorname{rank}(G)$ denotes the rank of the gain graph.

Flexible 2-periodic placement characterization. There exists a full placement-lattice $(p,L)$ of $G$ in $\mathbb{R}^2$ such that $(G,p,L)$ is a flexible full $2$-periodic framework if and only if $G$ has a type 1 flexible 2-lattice NBAC-colouring, a type 2 flexible 2-lattice NBAC-colouring, a type 3 flexible 2-lattice NBAC-colouring, a fixed lattice NBAC-colouring, or $\operatorname{rank}(G)<2$.

This conjecture would give a full characterization of flexible full $2$-periodic placements in terms of NBAC-colourings. The characterization is known for the other cases discussed in the paper, while the type 3 case is unresolved.