Weak chord-arc curves and double-dome quasisymmetric spheres

Let $\mathrm{\Omega }$ be a planar Jordan domain and $\alpha >0$. We consider double-dome-like surfaces $\mathrm{\Sigma }(\mathrm{\Omega },t^{\alpha })$ over $\bar{\mathrm{\Omega }}$ where the height of the surface over any point $x\in \bar{\mathrm{\Omega }}$ equals $\text{dist}(x,\mathrm{\partial }\mathrm{\Omega }{)}^{\alpha }$. We identify the necessary and sufficient conditions in terms of $\mathrm{\Omega }$ and $\alpha$ so that these surfaces are quasisymmetric to ${\mathbb{S}}^2$ and we show that $\mathrm{\Sigma }(\mathrm{\Omega },t^{\alpha })$ is quasisymmetric to the unit sphere ${\mathbb{S}}^2$ if and only if it is linearly locally connected and Ahlfors $2$-regular.