  The insulating magnetic pyrochlore Ce$_2$Zr$_2$O$_7$ has attracted much
attention as a quantum spin ice candidate with dipole-octupole character that
permits spin ice phases based not only on magnetic dipole moments but also
allows for even-more-exotic octupole-based spin ice phases. This work reports
low-temperature neutron diffraction measurements on single crystal
Ce$_2$Zr$_2$O$_7$ with $Q$-coverage both at low $Q$ where the magnetic form
factor for dipoles is near maximal and at high $Q$ covering the region where
the magnetic form factor for Ce$^{3+}$ octupoles is near maximal. This study
was motivated by recent powder neutron diffraction studies of other Ce-based
dipole-octupole pyrochlores, Ce$_2$Sn$_2$O$_7$ and Ce$_2$Hf$_2$O$_7$, which
each showed temperature-dependent diffuse diffraction at high $Q$ that was
interpreted as arising from octupolar correlations. Our measurements use an
optimized single crystal diffuse scattering instrument that allows us to screen
against strong single crystal Bragg scattering in Ce$_2$Zr$_2$O$_7$. The
temperature-difference neutron diffraction reveals a low-$Q$ peak consistent
with dipolar spin ice correlations. For larger $Q$, the temperature-difference
neutron diffraction shows an alternation between positive and negative net
intensity. These features are qualitatively consistent with the corresponding
numerical-linked-cluster (NLC) calculations using pseudospin interaction
parameters reported for Ce$_2$Zr$_2$O$_7$, Ce$_2$Sn$_2$O$_7$, or
Ce$_2$Hf$_2$O$_7$. Importantly, neither the measured data nor any of the NLC
calculations show increased scattering at high $Q$ resulting from octupolar
correlations. We conclude that at the lowest attainable temperatures for our
measurement ($T = 0.05$ K), octupolar correlations are not present in
Ce$_2$Zr$_2$O$_7$ on the level of our observation threshold of $\sim$ 0.1$\%$
of the low-$Q$ dipole scattering.
