Huang et al.'s boundary uniqueness conjecture for holomorphic mappings

Let $\Delta^+$ be the upper half disc in $\mathbb C$. Let $\Gamma_C=\{z\in\mathbb C:|\operatorname{Im}(z)|\leq C|\operatorname{Re}(z)|\}$ for some positive $C$, and let $f$ be a holomorphic function on $\Delta^+$ with continuous extension up to $(-1,1)$. Suppose that $f$ maps $(-1,1)$ into $\Gamma_C$ and vanishes to infinite order at $0$, meaning that for every $N\in\mathbb N$,

$$\lim_{\Delta^+\ni z\to 0}\frac{f(z)}{|z|^N}=0.$$

Huang et al.'s conjecture. Under these assumptions, $f$ vanishes identically.

The conjecture concerns boundary unique continuation for holomorphic functions whose boundary values lie in a cone. The supplied source does not indicate whether the conjecture has been resolved.