The Hardy-order conjecture for harmonic K-quasiconformal mappings

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

$$\sup_{f\in\mathcal{S}_\mathcal{H}(K)}|a_2|=\frac{3K+1}{K+1},$$

and, in particular,

$$f\in h^p\qquad\left(0<p<\frac{1}{2K}\right).$$

The bound for the Hardy-space order is expected to be sharp; proving this together with Nowak's conjecture would settle the cited Pavlović open problem completely.