Uniform distortion conjecture for simplicial power maps

Let $T=\triangle(,0,)$ be a triangle and denote its angle at $0$ by $\theta=\measuredangle(\xi,0,\eta)$. Let $T^q$ be the $q$th level of the regular 1-4 subdivision of $T$. For $\gamma>0$, define the power map $h(z)=z^\gamma$, and assume that $\gamma\theta<\pi$. Uniform distortion conjecture. The simplicial maps $h^q$, obtained by sampling $h$ at the vertices of $T^q$ and extending by linearity, are homeomorphisms satisfying

$$\mathrm{\textbf{D}}(h^q)\leq K$$

for some $K\geq 1$ independent of $q$. This conjecture concerns uniform quasiconformal control of simplicial approximations to power maps; the stated claim is not accompanied by a resolution in the supplied text.