Embeddedness conjecture for hexagonal circle patterns approximating power maps

Let $0<\alpha=\beta<\frac{1}{2}$ and set $\gamma=1-2\alpha$, so that $0<\gamma<1$. Consider the hexagonal circle patterns $z^{3\alpha}$ with an intersection point at the origin and $z^{3\gamma}$ with a circle at the origin, constructed with $\theta=2\pi\alpha$. Embeddedness conjecture. For $0<\alpha<\frac{1}{2}$, both of these hexagonal circle patterns are embedded. This asserts regular geometric behavior for the discrete analogues of the power maps $z\mapsto z^{3\alpha}$ and $z\mapsto z^{3\gamma}$, contrasting with the non-embedded patterns that can arise from other initial values; the supplied text gives no resolution of the claim.