The covering-radius conjecture for harmonic K-quasiconformal mappings

For $0<r<1$, define the disk

$$\mathbb{D}_r:=\{z\in\mathbb{C}:|z|<r\},$$

and let $K\geq1$ and $f\in\mathcal{S}^0_\mathcal{H}(K)$. Define

$$R:=\sup_{0<r<1}\{r:\mathbb{D}_r\subseteq f(\mathbb{D}),\ f\in\mathcal{S}^0_\mathcal{H}(K)\}.$$

Covering-radius conjecture. The radius satisfies

$$R\geq\frac{K+1}{6K+2}.$$

The radius is expected to be sharp, and the source presents this as an open conjecture associated with a covering theorem.