Pommerenke's integral conjecture for univalent meromorphic functions

Let $\Sigma_0$ be the class of univalent meromorphic functions $F$ in the exterior of the unit disk, normalized by $F(z)=z+O(z^{-1})$ as $z\to\infty$. Let $K=F(\mathbb{B}^c)^c$ contain $0$, and let $F_0(z)=z+z^{-1}$. Pommerenke's conjecture.

$$\frac{1}{2\pi}\int_{-\pi}^{\pi}|F(e^{i\theta})|\,d\theta\leq\frac{1}{2\pi}\int_{-\pi}^{\pi}|F_0(e^{i\theta})|\,d\theta=\frac{4}{\pi}.$$

The source reports the best known upper bound as $4.02/\pi$ and asks whether it can be improved to $4/\pi$; hence the conjecture remains open there.