  We present the evolution of Ly$\alpha$ emission derived from 53 galaxies at
$z=6.6-13.2$ that are identified by multiple JWST/NIRSpec spectroscopy programs
of ERS, GO, DDT, and GTO. These galaxies fall on the star-formation main
sequence and are the typical star-forming galaxies with UV magnitudes of
$-22.5\leq M_\mathrm{UV}\leq-17.0$. We find that 15 out of 53 galaxies show
Ly$\alpha$ emission at the $>3\sigma$ levels, and obtain Ly$\alpha$ equivalent
width (EW) measurements and stringent $3\sigma$ upper limits for the 15 and 38
galaxies, respectively. Confirming that Ly$\alpha$ velocity offsets and line
widths of our galaxies are comparable with those of low-redshift Ly$\alpha$
emitters, we investigate the redshift evolution of the Ly$\alpha$ EW. We find
that Ly$\alpha$ EWs statistically decrease towards high redshifts on the
Ly$\alpha$ EW vs. $M_{\rm UV}$ plane for various probability distributions of
the uncertainties. We then evaluate neutral hydrogen fractions $x_{\rm HI}$
with the Ly$\alpha$ EW redshift evolution and the cosmic reionization
simulation results on the basis of a Bayesian inference framework, and obtain
$x_{\rm HI}<0.79$, $=0.62^{+0.15}_{-0.36}$, and $0.93^{+0.04}_{-0.07}$ at
$z\sim7$, $8$, and $9-13$, respectively. These moderately large $x_{\rm HI}$
values are consistent with the Planck CMB optical depth measurement and
previous $x_{\rm HI}$ constraints from galaxy and QSO Ly$\alpha$ damping wing
absorptions, and strongly indicate a late reionization history. Such a late
reionization history suggests that major sources of reionization would emerge
late and be hosted by moderately massive halos in contrast with the
widely-accepted picture of abundant low-mass objects for the sources of
reionization.
