  OTOC has been used to characterize the information scrambling in quantum
systems. Recent studies showed that local conserved quantities play a crucial
role in governing the relaxation dynamics of OTOC in non-integrable systems. In
particular, slow scrambling of OTOC is seen for observables that has an overlap
with local conserved quantities. However, an observable may not overlap with
the Hamiltonian, but with the Hamiltonian elevated to an exponent larger than
one. Here, we show that higher exponents correspond to faster relaxation,
although still algebraic, and with exponents that can increase indefinitely.
Our analytical results are supported by numerical experiments.
