The tilt conjecture for the hat tiling growth form

Let $\alpha$ satisfy

$$\tan(\alpha)=\frac{\sqrt{3}}{3+2\tau},$$

where $\tau$ is the golden ratio, and let $\gamma$ denote the tilt angle of the growth form of the hat tiling. Hat growth-form tilt conjecture. The growth form of the hat tiling is tilted by

$$\gamma=-\alpha=-0.270919.$$

The conjecture is motivated by the slopes of the triangulation lines and by computer experiments. The paper does not prove the asserted tilt angle.