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.