  We present a realistic supersymmetric $\mu$-hybrid inflation model within the
framework of $SU(4)_C \times SU(2)_L \times U(1)_R$ gauge symmetry, wherein the
symmetry breaking $SU(4)_C \times SU(2)_L \times U(1)_R\rightarrow
SU(3)_C\times SU(2)_L \times U(1)_{B-L}\times U(1)_R$ occurs before observable
inflation, effectively eliminating topologically stable primordial monopoles.
Subsequent breaking of $U(1)_{B-L} \times U(1)_R \rightarrow U(1)_Y$ after
inflation leads to the formation of superheavy metastable cosmic strings (CSs),
capable of producing a stochastic gravitational wave background (SGWB)
consistent with the recent PTA data. Moreover, the scalar spectral index $n_s$
and the tensor-to-scalar ratio $r$ align with Planck 2018 observations. A
consistent scenario for reheating and non-thermal leptogenesis is employed to
explain the observed matter content of the universe. Finally, the embedding of
$G_{421}$ into the Pati-Salam gauge symmetry $G_{422}$ is briefly discussed,
predicting potentially observable proton decay rates detectable at facilities
such as Hyper Kamiokande and DUNE.
