Coefficient bound conjecture for the class of analytic functions associated with sine hyperbolic functions

Let $f\in\mathcal{G}_{sh}$ be an analytic function of the form given by equation (1), with Taylor coefficients $a_n$. Coefficient bound conjecture. One should have

$$|a_n|\leq\frac{1}{n-1}\qquad\text{for }n\geq 2.$$

The preceding argument establishes the bounds for $n=2,3,4,5$; the asserted estimate for all $n\geq 2$ remains unproved in the supplied text.