  Topological order offers possibilities for processing quantum information
which can be immune to imperfections. However, the question of its stability
out of equilibrium is relevant for experiments, where coupling to an
environment is unavoidable. In this work we demonstrate the robustness of
certain aspects of $Z_2 \times Z_2$ symmetry-protected topological (SPT) order
against a wide class of dissipation channels in the Lindblad and quantum
trajectory formalisms of an open quantum system. This is illustrated using the
one-dimensional $ZXZ$ cluster Hamiltonian along with Pauli-string jump
operators. We show that certain choices of dissipation retaining strong
symmetries support a steady-state manifold consisting of two non-local logical
qubits, and for Hamiltonian perturbations preserving the global symmetry, the
manifold remains long-lived. In contrast, this metastability is destroyed upon
breaking the above-mentioned symmetry. While the localized edge qubits of the
cluster Hamiltonian are not conserved by the Lindbladian evolution, they do
correspond to weak symmetries and thus retain a memory of their initial state
at all times in the quantum trajectories. We utilize this feature to construct
protocols to retrieve the quantum information either by monitoring jumps or
error mitigation. Our work thus proposes a novel framework to study the
dynamics of dissipative SPT phases and opens the possibility of engineering
entangled states relevant to quantum information processing.
