The area conjecture for growth forms of deformed hat tilings

For positive $b\neq 1$, let $Tile(1,b)$ be the deformed hat tile, and let a tiling by these tiles have growth form with area $area_{growth\ form}$. The tile has area $area_{Tile(1,b)}$. Area conjecture for deformed hat tilings. The growth forms resulting from $Tile(1,b)$ satisfy

$$area_{growth\ form}=2\sqrt{3}\,area_{Tile(1,b)}.$$

This is proposed on the basis of computer experiments and the close relationship between hat tiling and the hexagonal lattice. The corresponding tilings are stated to be aperiodic and combinatorially equivalent, and their growth forms are known in the paper to be regular hexagons, but the asserted area relation is not proved there.