Let $(X^{4}=M\setminus D,\omega)$ be given by the Tian-Yau construction where $M$ is a smooth Fano manifold of complex dimension $2$ and $D$ is a smooth anti-canonical divisor. Then there is some positive constant $\delta_{h}>0$ which depends only on the geometry of $X^{4}$ such that if a 1-form $\gamma$ on $X$ satisfies

(5.1)

$$d^{+}\gamma=d^{*}\gamma=0$$

and

(5.2)

$$|\gamma|_{\omega}=O(e^{\delta_{h}r^{\frac{2}{3}}})$$

as $r\to\infty$, where $r$ is the distance function on $X$ to a fixed point, then $\gamma=0$.