Improved coefficient conjecture for univalent harmonic mappings

Let $f=h+\overline{g}\in\mathcal{S}_{\mathcal{H}}$, where $h(z)=z+\sum_{n=2}^{\infty}a_nz^n$ and $g(z)=\sum_{n=1}^{\infty}b_nz^n$ are the analytic and co-analytic parts of a normalized univalent harmonic mapping. Improved coefficient conjecture. For a parameter $a$ with $-1<a<1$, the coefficients satisfy

$$|a_n|\leq \frac{2(1+a)n^2+3(1-a)n+(1+a)}{6},\qquad |b_n|\leq \frac{2(1+a)n^2+3(a-1)n+(1+a)}{6}$$

for all $n\geq2$. The paper proposes these bounds as an improved form of the earlier conjecture, motivated by generalized harmonic Koebe functions and their coefficient estimates; their validity for the full class remains the stated conjectural issue.