  Relativistic winds of fast-spinning pulsars have been proposed as a potential
site for cosmic-ray acceleration from very high energies (VHE) to ultrahigh
energies (UHE). We re-examine conditions for high-energy neutrino production,
considering the interaction of accelerated particles with baryons of the
expanding supernova ejecta and the radiation fields in the wind nebula. We make
use of the current IceCube sensitivity in diffusive high-energy neutrino
background, in order to constrain the parameter space of the most extreme
neutron stars as sources of VHE and UHE cosmic rays. We demonstrate that the
current non-observation of $10^{18}$ eV neutrinos put stringent constraints on
the pulsar scenario. For a given model, birthrates, ejecta mass and
acceleration efficiency of the magnetar sources can be constrained. When we
assume a proton cosmic ray composition and spherical supernovae ejecta, we find
that the IceCube limits almost exclude their significant contribution to the
observed UHE cosmic-ray flux. Furthermore, we consider scenarios where a
fraction of cosmic rays can escape from jet-like structures piercing the
ejecta, without significant interactions. Such scenarios would enable the
production of UHE cosmic rays and help remove the tension between their EeV
neutrino production and the observational data.
