  We prove that all eigenstates of many-body localized symmetry protected
topological systems with time reversal symmetry have four-fold degenerate
entanglement spectra in the thermodynamic limit. To that end, we employ unitary
quantum circuits where the number of sites the gates act on grows linearly with
the system size. We find that the corresponding matrix product operator
representation has similar local symmetries as matrix product ground states of
symmetry protected topological phases. Those local symmetries give rise to a
$\mathbb{Z}_2$ topological index, which is robust against arbitrary
perturbations so long as they do not break time reversal symmetry or drive the
system out of the fully many-body localized phase.
