Log-harmonic coefficient conjecture

Let $f(z)=zh(z)\overline{g(z)}\in\mathcal{S}_{Lh}$, where $h$ and $g$ are given by the coefficient expansions in the source. Thus, for $n\geq 1$, $a_n$ and $b_n$ denote the corresponding analytic and co-analytic coefficients. Log-harmonic coefficient conjecture. For all $n\geq 1$,

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

This is proposed as an analogue of the analytic and harmonic Bieberbach conjectures. It is known for starlike log-harmonic mappings; the general case remains open.