  We present time-resolved and CW optical spectroscopy studies of interlayer
excitons (IXs) in 2$H-$ and 3$R-$stacked MoSe$_2$/WSe$_2$ heterobilayers and
obtain evidence for the strong participation of hot phonons in the underlying
photo-physics. Photoluminescence excitation spectroscopy reveals that excess
energy associated with optical excitation of \textit{intra}-layer excitons and
relaxation to IXs affects the overall IX-PL lineshape, while the spectrally
narrow emission lines conventionally associated with moir\'e IXs are
unaffected. A striking uniform line-spacing of the sharp emission lines is
observed together with temperature and excitation level dependent spectra
suggesting an entirely new picture that photo-generated phonons lead to
phonon-replicas shaping the IX-emission. Excitation power and time resolved
data indicate that these features are polaronic in nature. Our experimental
findings modify our current understanding of the photophysics of IXs beyond
current interpretations based on moir\'e-trapped IXs.
