  Delafossite PdCrO$_2$ is an intriguing material which displays nearly-free
electron and Mott insulating behaviour in different layers. Both angle-resolved
photoemission spectroscopy (ARPES) and Compton scattering measurements have
established a hexagonal Fermi surface in the material's paramagnetic phase.
However, the Compton experiment detected an additional structure in the
projected occupancy which was originally interpreted as an additional Fermi
surface feature not seen by ARPES. Here, we revisit this interpretation of the
Compton data. State-of-the-art density functional theory (DFT) with dynamical
mean field theory (DMFT), the so-called DFT+DMFT method, predicts the Mott
insulating state along with a single hexagonal Fermi surface in excellent
agreement with ARPES and Compton. However, DFT+DMFT fails to predict the
intensity of the additional spectral weight feature observed in the Compton
data. We infer that this discrepancy may arise from the DFT+DMFT not being able
to correctly predict certain features in the shape and dispersion of the
unoccupied quasiparticle band near the Fermi level. Therefore, a theoretical
description beyond our DFT+DMFT model is needed to incorporate vital electron
interactions, such as inter-layer electron coupling interactions which for
PdCrO$_2$ gives rise to the Kondo-like so-called intertwined excitation.
