Clunie–Sheil-Small coefficient conjecture in the full harmonic class

Prove the sharp coefficient estimates |a_n|−|b_n|≤n, |a_n|≤(n+1)(2n+1)/6, and |b_n|≤(n−1)(2n−1)/6 for every normalized sense-preserving univalent harmonic mapping f=h+g in S_H^0 and every integer n≥2.

Background

The conjecture is presented as the harmonic analogue of the Bieberbach conjecture. It specifies sharp bounds for the analytic and co-analytic Taylor coefficients of normalized univalent harmonic mappings, with equality attained by the harmonic Koebe function.

The estimates have been proved for several geometric subclasses, including starlike, typically real, close-to-convex, and convex-in-one-direction mappings. The paper explicitly states that the conjecture is still unresolved for the entire class S_H^0.