Coefficient conjecture for K-quasiconformal harmonic mappings

Let $f=h+\overline{g}\in\mathcal{S}^0_H(K)$, with $h$ and $g$ having the series representations specified in the source. Let $A(n,k)$ and $B(n,k)$ be the quantities defined in the source's equations (eq-mak) and (eq-mbk), respectively, and let $P_k$ be the function defined in equation (eq-qhk). Coefficient conjecture for K-quasiconformal harmonic mappings. For $n=2,3,\ldots$,

$$|a_n|\leq A(n,k),\qquad |b_n|\leq B(n,k),\qquad \big||a_n|-|b_n|\big|\leq n.$$

Equalities occur for the function $P_k$. The source attributes this conjecture to the authors cited as wwrq and notes that it was proposed without much supporting evidence; its resolution is not given.