A sharp inequality for the order of the minimal positive harmonic function in $T$-homogeneous domain

Let $G$ be a simply connected domain in $\mathbb C$ which is $T$-homoheneous, i.e., $TG=G$ for some $T>0$. Let $\rho(G)$ be the order of the minimal positive harmonic function in $G$. We prove that a kind of symmetrization of $G$ and prove that it does not increase $\rho(G)$. This implies a sharp lower bound for $\rho(G)$ in terms of conformal modulus of a quadrilateral naturally connected with $G$.