  In this paper, we prove Huang et al.'s conjecture stated that if $f$ is a
holomorphic function on $\Delta^+:=\{z\in \mathbb C \colon
|z|<1,~\mathrm{Im}(z)>0\}$ with $\mathcal{C}^\infty$-smooth extension up to
$(-1,1)$ such that $f$ maps $(-1,1)$ into a cone $\Gamma_C:=\{z\in \mathbb
C\colon |\mathrm{Im} (z)| \leq C|\mathrm{Re} (z)|\}$, for some positive number
$C$, and $f$ vanishes to infinite order at $0$, then $f$ vanishes identically.
In addition, some regularity properties of the Riemann mapping functions on the
boundary and an example concerning Huang et al.'s conjecture are also given.
