  Recent investigations on the dipolar-octupolar compounds Ce$_2$Zr$_2$O$_7$
and Ce$_2$Sn$_2$O$_7$ suggest that they may stabilize so-called $\pi$-flux
octupolar quantum spin ice ($\pi$-O-QSI), a novel three-dimensional quantum
spin liquid hosting emergent photons. Confirmation of such an exotic phase
would require the prediction of a distinctive signature and its subsequent
experimental observation. So far, however, theoretical predictions for any such
sharp smoking-gun signatures are lacking. In this Letter, we thoroughly
investigate O-QSI using an extension of gauge mean-field theory. This framework
produces a phase diagram consistent with previous studies and an
energy-integrated neutron scattering signal with intensity-modulated rod
motifs, as reported in experiments and numerical studies. We predict that the
dynamical spin structure factor of $\pi$-O-QSI is characterized by a broad
continuum with three distinctive peaks as a consequence of the two mostly flat
spinon bands. These three peaks should be measurable by high-resolution
inelastic neutron scattering. Such spectroscopic signatures would be clear
evidence for the realization of $\pi$-flux quantum spin ice.
