Conjectured covering radius for K-quasiconformal harmonic mappings

Let $\mathcal{S}^0_H(K)$ denote the class used in the source of normalized $K$-quasiconformal harmonic mappings, and let $k$ be the corresponding parameter. Define

$$M(k)=\frac{k^2+8k-1}{4(1-k)^2}+\frac{2k(k+1)}{(1-k)^3}\log\frac{1+k}{2}.$$

Covering-radius conjecture. Every function $f\in\mathcal{S}^0_H(K)$ contains the disk

$$\{w:\ |w|<M(k)\}.$$

The proposed radius satisfies $\lim_{k\to1^-}M(k)=-\frac16$ in the preceding calculation, and the source presents the assertion as a natural proposal without establishing its validity.