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.