Reverse Harnack conjecture for non-null-homotopic harmonic mappings

Let $\Omega\subset\mathbb C$ be a doubly connected domain, and let $\mathbb C_\circ=\mathbb C\setminus\{0\}$. Suppose $h\colon\Omega\to\mathbb C_\circ$ is harmonic and is not homotopic to a constant through continuous mappings from $\Omega$ to $\mathbb C_\circ$. Reverse Harnack conjecture. Then

$$\frac{\sup_{\Omega}|h|}{\inf_{\Omega}|h|}\geqslant\cosh\left(\frac{1}{2}\operatorname{Mod}\Omega\right).$$

If $h$ is additionally injective, the factor $1/2$ should be omitted. This would extend the preceding sharp reverse Harnack estimate by removing the assumption that a boundary component is mapped homeomorphically onto the unit circle; the given text does not report a resolution.