Sharp coefficient bounds for Gregory-starlike functions

Let $f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots$ belong to the class $\mathcal{S}_{G}^{\ast}$. The coefficients $a_n$ are the Taylor coefficients of $f$. Coefficient-bound conjecture. The sixth coefficient satisfies

$$\left\vert a_{6}\right\vert \leq \frac{1}{10},$$

and, more generally, the sharp inequality

$$\left\vert a_{n}\right\vert \leq \frac{1}{2(n-1)}$$

holds for all $n\in\mathbb{N}\backslash\{1\}$. The first five coefficient bounds are established as sharp, while sharpness of the sixth bound and the general inequality remain open questions.