In arXiv:2103.15482 an estimate for suitable skew-symmetric 2-tensors was
    claimed. Soon after, this estimate has been exploited to claim powerful
    classification results: most notably, it has been employed to propose a proof
    of a Black Hole Uniqueness Theorem for vacuum static spacetimes with positive
    scalar curvature and in connection with the Besse Conjecture. In the present
    note we point out an issue in the argument proposed in arXiv:2103.15482 and we
    provide a counterexample to the estimate.

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