= Solution
The forms $dz_I\wedge d\overline z_J$ are parallel for the flat metric. Consequently the Hodge Laplacian acts coefficientwise:
$$
\Delta\alpha=\sum_{I,J}(\Delta\alpha_{I,J}),dz_I\wedge d\overline z_J.
$$
If every coefficient is constant, this vanishes. Conversely, if $\Delta\alpha=0$, orthogonality of the constant frame gives $\Delta\alpha_{I,J}=0$ for every $I,J$. Each coefficient is a harmonic function on a compact connected manifold and hence is constant by the <Strong maximum principle for harmonic functions>.
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