  We consider the following conjecture (from Huang, et al): Let $\Delta^+$
denote the upper half disc in $\mathbb{C}$ and let $\gamma = ( - 1, 1)$ (viewed
as an interval in the real axis in $\mathbb{C}$). Assume that $F$ is a
holomorphic function on $\Delta^+$ with continuous extension up to $\gamma$
such that $F$ maps $\gamma$ into $\{|\mbox{Im} z|\leq C|\mbox{Re} z|\},$ for
some positive $C.$ If $F$ vanishes to infinite order at $0$ then $F$ vanishes
identically. We show that given the conditions of the conjecture, either
$F\equiv 0$ or there is a sequence in $\Delta^+$, converging to $0,$ along
which $\mbox{Im} F/\mbox{Re} F$ (defined where $\mbox{Re} F\neq 0$) is
unbounded.
