  We study the case of $\mathcal{PT}$-symmetric perturbations of Hermitian
Hamiltonians with degenerate eigenvalues using the example of a triple-well
system. The degeneracy complicates the question, whether or not a stationary
current through such a system can be established, i.e.\ whether or not the
$\mathcal{PT}$-symmetric states are stable. It is shown that this is only the
case for perturbations that do not couple to any of the degenerate states. The
physical explanation for the inhibition of stable currents is discussed.
However, introducing an on-site interaction restores the capability to support
stable currents.
