The sharp second-coefficient conjecture for harmonic K-quasiconformal mappings

Let $\mathcal{S}^0_\mathcal{H}(K)$ be the class of normalized harmonic $K$-quasiconformal mappings, and write $f=h+\overline{g}$. Second-coefficient conjecture. One has

$$\sup_{f\in\mathcal{S}^0_\mathcal{H}(K)}|a_2|=\frac{5K+3}{2K+2}.$$

The bound is expected to be sharp and is motivated by the HQC Bieberbach conjecture; the source does not report a proof.