Sharp modulus inequality for energy-minimal diffeomorphisms

Let $\Omega$ and $\Omega^*$ be bounded doubly connected domains in $\mathbb{C}$, and suppose that there is an energy-minimal diffeomorphism

$$h\colon \Omega\overset{{}_{\textnormal{\tiny{onto}}}}{\longrightarrow}\Omega^*.$$

Write $\operatorname{Mod}\Omega$ and $\operatorname{Mod}\Omega^*$ for their moduli. Modulus conjecture. One has

$$\operatorname{Mod}\Omega^*\geqslant \log\cosh \operatorname{Mod}\Omega.$$

Moreover, if both sides are finite and equal, then $\Omega^*$ is a circular annulus. This is proposed as a sharp form of the preceding existence theorem; the equality case predicts the precise extremal geometry, while the general assertion remains open.