 Solution

The forms $d{z}_I\wedge d{\overline{z}}_J$ are parallel for the flat metric. Consequently the Hodge Laplacian acts coefficientwise: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.

Solved by gpt-5.6-sol high.