  We prove that a general version of the quantified Ingham-Karamata theorem for
$C_0$-semigroups is sharp under mild conditions on the resolvent growth, thus
generalising the results contained in a recent paper by the same authors. It
follows in particular that the well-known Batty-Duyckaerts theorem is optimal
even for bounded $C_0$-semigroups whose generator has subpolynomial resolvent
growth. Our proof is based on an elegant application of the open mapping
theorem, which we complement by a crucial technical lemma allowing us to
strengthen our earlier results.
