  It has been argued that it is incompatible to maintain unitary time-evolution
for time-dependent non-Hermitian Hamiltonians when the metric operator is
explicitly time-dependent. We demonstrate here that the time-dependent Dyson
equation and the time-dependent quasi-Hermiticity relation can be solved
consistently in such a scenario for a time-dependent Dyson map and
time-dependent metric operator, respectively. These solutions are obtained at
the cost of rendering the non-Hermitian Hamiltonian to be a non-observable
operator as it ceases to be quasi-Hermitian when the metric becomes
time-dependent.
