I concur. There is no way a mechanical lift pump problem would have allowed it to run better when it warms up.
The 1994-2001 DS4 6.5L uses an electric lift pump mounted on the frame under the driver seat.
The lift pump might've been the only problem but I've seen similar operation with injection pump solenoid issues. Properly rebuilt DS4 pumps are not cheap and they're a pain in the neck to replace.
The DS4 isn't oogy boogie or a dark art or whatever other comment that can be made about it. They require a little more care than the old DB2 on the 93 & prior 6.5L & 6.2L engines. Again the snake oil salesmen that tell you you need to relocate the PMD are full of it. I've run five DS4 equipped trucks for over 300,000 miles without trouble from the PMD. I did remove the PMD, dress the mounting pad so it was flat and smooth and re-install them with heatsink grease.