  Motivated by recent experiments on Ce$_2$Zr$_2$O$_7$ that reveal a dynamic,
liquid-like ground state, we study the nearest neighbor XYZ Hamiltonian of
dipolar-octupolar pyrochlore magnets with the pseudofermion functional
renormalization group (PFFRG), which is numerically implemented by the
SpinParser software. Taking the interaction between the octupolar components to
be dominant and antiferromagnetic, we map out the phase diagram demarcating the
quantum disordered and magnetically ordered states. We identify four distinct
phases, namely the $0$-flux and $\pi$-flux quantum spin ices, and the
all-in-all-out magnetic orders along the local $z$ and $x$ axes. We further use
the static two-spin correlations output by the PFFRG algorithm to compute the
polarized neutron scattering cross-sections, which are able to capture several
qualitative features observed experimentally, in the materially relevant
parameter regime that stabilizes the $\pi$-flux quantum spin ice. Our results
provide support for a quantum spin liquid ground state in Ce$_2$Zr$_2$O$_7$.
