  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 that 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 non-free. In another application, we show that an
irreducible multiarrangement of rank $3$ with at least two locally heavy
hyperplanes is not free.
