  Quantum metrology protocols allow to surpass precision limits typical to
classical statistics. However, in recent years, no-go theorems have been
formulated, which state that typical forms of uncorrelated noise can constrain
the quantum enhancement to a constant factor, and thus bound the error to the
standard asymptotic scaling. In particular, that is the case of
time-homogeneous (Lindbladian) dephasing and, more generally, all semigroup
dynamics that include phase covariant terms, which commute with the system
Hamiltonian. We show that the standard scaling can be surpassed when the
dynamics is no longer ruled by a semigroup and becomes time-inhomogeneous. In
this case, the ultimate precision is determined by the system short-time
behaviour, which when exhibiting the natural Zeno regime leads to a
non-standard asymptotic resolution. In particular, we demonstrate that the
relevant noise feature dictating the precision is the violation of the
semigroup property at short timescales, while non-Markovianity does not play
any specific role.
