  Neutrinos can be pseudo-Dirac in Nature - they can be Majorana fermions while
behaving effectively as Dirac fermions. Such scenarios predict active-sterile
neutrino oscillations driven by a tiny mass-squared difference $(\delta m^2)$,
which is an outcome of soft lepton number violation. Oscillations due to tiny
$\delta m^2$ can only take place over astrophysical baselines and hence are not
accessible in terrestrial neutrino oscillation experiments. This implies that
high-energy neutrinos coming from large distances can be naturally used to test
this scenario. We use the recent observation of high-energy neutrinos from the
active galactic nuclei NGC 1068 by the IceCube collaboration to rule out
$\delta m^2$ in the region $[1.4 \times 10^{-18}, 10^{-17}]\, {\rm eV}^2$ at
more than $90\%$ confidence level - one of the strongest limits to date on the
values of $\delta m^2$. We also discuss possible uncertainties which can reduce
the sensitivity of these results.
