Clunie–Sheil-Small coefficient conjecture for normalized harmonic univalent mappings

Let $f=h+\overline{g}\in\mathcal{S}^0_H$, where $h$ and $g$ have the series representations specified in the source. The harmonic Koebe function is

$$K(z)=\frac{z-\frac{1}{2}z^2+\frac{1}{6}z^3}{(1-z)^3}+\overline{\left(\frac{\frac{1}{2}z^2+\frac{1}{6}z^3}{(1-z)^3}\right)}.$$

Clunie–Sheil-Small's coefficient conjecture. For every $n\geq 2$,

$$|a_n|\leq\frac{(n+1)(2n+1)}{6},\qquad |b_n|\leq\frac{(n-1)(2n-1)}{6},\qquad \big||a_n|-|b_n|\big|\leq n.$$

The bounds are attained by the harmonic Koebe function $K$. This is an open coefficient problem for harmonic univalent mappings, attributed in the source to Clunie and Sheil-Small.