  The Bernevig-Hughes-Zhang (BHZ) model, which serves as a cornerstone in the
study of the quantum spin Hall insulators, showcases robust spin-filtered
helical edge states in a nanoribbon geometry. In the presence of an in-plane
magnetic field, these (first-order) helical states gap out to be replaced by
second-order corner states under suitable open-boundary conditions. Here, we
show that the inclusion of a spin-dependent non-Hermitian balanced gain/loss
potential induces a competition between these first and second-order
topological phases. Surprisingly, the previously dormant first-order helical
edge states in the nanoribbon resurface as the non-Hermitian effect
intensifies, effectively neutralizing the role played by the magnetic field. By
employing the projected spin spectra and the spin Chern number, we conclusively
explain the resurgence of the first-order topological properties in the
time-reversal symmetry-broken BHZ model in presence of non-Hermiticity.
Finally, the biorthogonal spin-resolved Berry phase, exhibiting a non-trivial
winding, definitively establishes the topological nature of these revived edge
states, emphasizing the dominance of non-Hermiticity over the magnetic field.
