  It is well known that non-Hermitian systems can be extremely sensitive to
boundary conditions owing to non-Hermitian skin effect (NHSE). Analogously, we
investigate two boundary-sensitive $U(1)$ symmetric Lindbladians: one carries
current in the steady state, and the other does not. The numerical results
indicate significant change of the Liouvillian spectrum, eigenmodes and
relaxation time for both Lindbladians when the boundary conditions are altered.
This phenomenon is found to be triggered by the Liouvillian skin effect (LSE),
specifically the localization of eigenmodes, which stems from the NHSE of the
non-Hermitian effective Hamiltonian. In addition, these two Lindbladians
manifest different LSE, ultimately resulting in distinct relaxation behaviors.
