The signed-tiling conjecture for contributions of regions in the double dimer model

Let $L$ be a region in the hexagonal lattice $H$, and let its contribution be the connection around $L$ in the model. A bone is the union of three collinear adjacent hexagons, a stone is the union of three hexagons that share a common vertex, and a snake is a union of four hexagons in an S shape. A signed tiling of $L$ is a collection of these tiles, each assigned weight $+1$ or $-1$, such that the total contribution at every hexagon inside $L$ is $1$ and at every hexagon outside $L$ is $0$. Let $I$ denote the $2\times2$ identity matrix.

Signed-tiling conjecture. If $n=3$ or $n=6$, then the contribution of $L$ is $0$ unless there exists a signed tiling of $L$ by stones, bones, and snakes; if such a signed tiling exists, the contribution is

$$(-1)^s\cdot I,$$

where $s$ is the number of stones used in the signed tiling.

This conjecture gives a combinatorial criterion for when the connection associated with a region is nonzero at the specified roots of unity, and determines the resulting matrix from the number of stones in a signed tiling. The supplied text does not state whether the claim has been proved or remains open.