  We present a successful realization of supersymmetric $\mu$-hybrid inflation
model based on a gauged $U(1)_{B-L}$ extension of the minimal supersymmetric
standard model, with the soft supersymmetry breaking terms are playing an
important role. Successful non-thermal leptogenesis with gravitino dark matter
yields a reheat temperature in the range $2 \times 10^{7} \lesssim T_R \lesssim
5 \times 10^{9}$ GeV. This corresponds to the predictions $2 \times 10^{-18}
\lesssim r\lesssim 4 \times 10^{-13}$ for the tensor to scalar ratio, and $-2
\times 10^{-6} \lesssim dn_s/d\ln k \lesssim -5 \times 10^{-11}$ for the
running of the scalar spectral index. The $B-L$ breaking scale is estimated as
$ 6 \times 10^{14}\lesssim M/ \text{GeV}\lesssim 10^{16}$, calculated at the
central value of the scalar spectral index, $n_s =0.9655$, reported by Planck
2018. Finally, in a grand unified theory setup the dimensionless string tension
parameter associated with the metastable strings is in the range $ 10^{-9}
\lesssim G\mu_\text{cs} \lesssim 10^{-6}$ corresponding to a stochastic
gravitational wave background lying within the 2$\sigma$ bound of the recent
NANOGrav 12.5-yr data.
