Hankel determinant bound for functions in the class U with vanishing second coefficient

Let $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$ belong to the class $\mathcal U$, and suppose that $a_2=0$. For an integer $n\ge3$, define the second Hankel determinant by

$$H_2(n)(f)=a_na_{n+2}-a_{n+1}^2.$$

Hankel determinant conjecture. For every integer $n\ge3$,

$$|H_2(n)(f)|=|a_na_{n+2}-a_{n+1}^2|\le1.$$

The estimate is sharp for $f(z)=\frac{z}{1-z^2}=\sum_{n=1}^{\infty}z^{2n-1}$. The result extends the sharp bounds established in the paper for the cases $n=2$ and $n=4$ to all higher second Hankel determinants for functions in $\mathcal U$ with missing second coefficient; the general case remains open.