Weak flexible 2-periodic placement characterization without type 3 colourings

Let $G$ be a connected $\mathbb{Z}^2$-gain graph. A full placement-lattice is a pair $(p,L)$ consisting of a placement and a full lattice in $\mathbb{R}^2$; the associated framework is flexible when it has a nontrivial continuous flex. The colouring types and $\operatorname{rank}(G)$ are as defined in the source.

Weak 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 fixed lattice NBAC-colouring, or

$$\operatorname{rank}(G)<2.$$

The paper presents this as the slightly stronger consequence of the type 3 exclusion conjecture, together with earlier lemmas. Its resolution is therefore not established in the supplied text.