  We discuss leptogenesis constraints on the mass of the right-handed $W$-boson
($W_R$) in a TeV-scale Left-Right seesaw model (LRSM) for neutrino masses. For
generic Dirac mass of the neutrinos, i.e. with all Yukawa couplings $\lesssim
10^{-5.5}$, it has been pointed out that successful leptogenesis requires a
lower bound of 18 TeV on the $W_R$ mass, pushing it beyond the reach of LHC.
Such TeV-scale LRSM must, however, be parity-asymmetric for type-I seesaw to
give the observed neutrino masses. This class of models can accommodate larger
Yukawa couplings, which give simultaneous fits to charged-lepton and neutrino
masses, by invoking either cancellations or specific symmetry-textures for
Dirac ($M_D$) and Majorana ($M_N$) masses in the seesaw formula. We show that
in this case, the leptogenesis bound on $M_{W_R}$ can be substantially weaker,
i.e. $M_{W_R}\gtrsim 3$ TeV for $M_N \lesssim M_{W_R}$. This happens due to
considerable reduction of the dilution effects from $W_R$-mediated decays and
scatterings, while the washout effects due to inverse decays are under control
for certain parameter ranges of the Yukawa couplings. We also show that this
model is consistent with all other low energy constraints, such as lepton
flavor violation and neutrinoless double beta decay. Thus, a discovery of the
right-handed gauge bosons alone at the LHC will not falsify leptogenesis as the
mechanism behind the matter-antimatter asymmetry in our Universe.
