Realization conjecture for type 3 flexible 2-lattice colourings

Let $G$ be a $\mathbb{Z}^2$-gain graph. A type 3 flexible 2-lattice NBAC-colouring is a NBAC-colouring of the type defined in the source, involving a nonzero $\alpha\in\mathbb{Z}^2$ and the stated span and almost-monochromatic-circuit conditions. A full placement-lattice is a placement and full lattice in $\mathbb{R}^2$, and a framework is flexible if it has a nontrivial continuous flex.

Type 3 realization conjecture. If $G$ has a type 3 flexible 2-lattice NBAC-colouring, then 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.

The paper explicitly identifies this as an open question. It would complete the realization theory for the remaining type of flexible $2$-lattice NBAC-colouring.