Conjecture on the transformed function in the class $\mathcal{U}$

Let $f\in\mathcal{U}$ have the normalization and coefficient $a_{2}$ used in the preceding theorem, and let $g$ be the function defined by expression (8). The notation $\mathcal{U}$ denotes the class under consideration.

Conjecture. The function $g$ belongs to the class $\mathcal{U}$ in the disc

$$|z|<\frac{|a_{2}|}{2}.$$

The result is best possible.

The authors state this as an expected extension of part (b) of the preceding theorem from the previously established range to all $0<|a_{2}|\leq 2$; the conjecture remains unresolved in the supplied text.