  The IceCube Collaboration has published four years of data and the observed
neutrino flux is significantly in excess of the expected atmospheric
background. Due to the steeply falling atmospheric background spectrum, events
at the highest energies are most likely extraterrestrial. In our previous
approach we have studied blazars as the possible origin of the High-Energy
Starting Events (HESE) neutrino events at PeV energies. In this work we extend
our study to include all HESE neutrinos (which does not include IC 170922A) at
or above a reconstructed energy of 100 TeV, but below 1 PeV. We study the X-ray
and $\gamma$-ray data of all ($\sim200$) 3LAC blazars that are positionally
consistent with the neutrino events above 100 TeV to determine the maximum
neutrino flux from these sources. This larger sample allows us to better
constrain the scaling factor between the observed and maximum number of
neutrino events. We find that when we consider a realistic neutrino spectrum
and other factors, the number of neutrinos is in good agreement with the
detected number of IceCube HESE events. We also show that there is no direct
correlation between \Fermi-LAT $\gamma$-ray flux and the IceCube neutrino flux
and that the expected number of neutrinos is consistent with the non-detection
of individual bright blazars.
