Stone-parity conjecture for signed tilings of the hexagonal grid

Let $R$ be a region in the hexagonal grid, and let $\partial R$ denote its boundary matrix product. A signed tiling of $R$ may use tiles of either weight, and stones are the tiles whose boundary contribution changes the sign. The minimum number of stones needed refers to the minimum over such signed tilings.

Stone-parity conjecture. The sign of $\partial R$ corresponds to the parity of the minimum number of stones needed for $R$ to be signed tilable when adding tiles of either weight along the boundary. Thus,

$$\partial R = (-1)^{\text{\#\{stones needed\}}}\cdot I.$$

The preceding results establish that $\partial R$ is either $I$ or $-I$ for a signed tilable region, and determine the sign when no stone or exactly one stone is required. The conjecture proposes the corresponding parity rule for the minimum number of stones in general.