  IceCube has reported the detection of a diffuse TeV-PeV neutrino emission,
for which the flat spectrum radio quasars (FSRQs) have been proposed to be the
candidate sources. Here we assume that the neutrino flux from FSRQs is
proportional to their gamma-ray ones, and obtain the gamma-ray/neutrino flux
ratio by the diffuse gamma-ray flux from Fermi-LAT measurement of FSRQs and the
diffuse neutrino flux detected by IceCube. We apply this ratio to individual
FSRQs and hence predict their neutrino flux. We find that a large fraction of
candidate FSRQs from the northern sky in the IceCube point source search has
predicted neutrino flux above the IceCube upper limit; and for the sample of
stacking search for neutrinos by IceCube, the predicted stacked flux is even
larger than the upper limit of stacked flux by orders of magnitude. Therefore
the IceCube limit from stacking searches, combined with the Fermi-LAT
observations, already rejects FSRQs as the main sources of IceCube-detected
diffuse neutrinos: FSRQs can only account for <10% (4%) of the IceCube-detected
diffuse neutrino flux, according to the stacking searches from the whole
(northern) sky. The derived neutrino/gamma-ray flux ratio is <<1, also implying
that the gamma-ray emission from FSRQs cannot be produced by the secondary
leptons and photons from the pion production processes. The caveat in the
assumptions is discussed.
