  The Kondo-Heinsberg chain is an interesting model of a strongly correlated
system which has a broad superconducting state with pair-density wave (PDW)
order. Some of us have recently proposed that this PDW state is a
symmetry-protected topological (SPT) state, and the gapped spin sector of the
model supports Majorana zero modes. In this work, we reexamine this problem
using a combination of numeric and analytic methods. In extensive density
matrix renormalization group calculations, we find no evidence of a topological
ground state degeneracy or the previously proposed Majorana zero modes in the
PDW phase of this model. This result motivated us to reexamine the original
arguments for the existence of the Majorana zero modes. A careful analysis of
the effective continuum field theory of the model shows that the Hilbert space
of the spin sector of the theory does not contain any single Majorana fermion
excitations. This analysis shows that the PDW state of the doped 1D
Kondo-Heisenberg model is not an SPT with Majorana zero modes.
