Hyperplane arrangements of rank 3 admitting an unbalanced Ziegler restriction are known to fulfill Terao's conjecture. This long-standing conjecture asks whether the freeness of an arrangement is determined by its combinatorics. In this note we prove that arrangements which admit a locally heavy flag satisfy Terao's conjecture which is a generalization of the statement above to arbitrary dimension. To this end we extend results characterizing the freeness of multiarrangements with a heavy hyperplane to those satisfying the weaker notion of a locally heavy hyperplane. As a corollary we give a new proof that irreducible arrangements with a generic hyperplane are totally nonfree. In another application we show that an irreducible multiarrangement of rank 3 with at least two locally heavy hyperplanes is not free.
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