= Solution
The <symmetry rule for extremal length> says that if a domain and its curve family are invariant under reflection, the supremum defining <extremal length> may be taken over reflection-invariant <conformal metrics>. More generally, for a conformal or anticonformal involution $T$, use metrics satisfying $T^*\rho=\rho$.
To see this, replace $\rho$ by $\rho_s=(\rho+T^*\rho)/2$. Every curve in the invariant family and its image have length at least $L_\rho$, so $L_{\rho_s}\geq L_\rho$. Convexity of the square and change of variables give $A_{\rho_s}\leq A_\rho$. Therefore symmetrizing cannot reduce the length-squared to area ratio.
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