My issue - seeing precisely this behavior - was all down to the infamous gunked-up-likely-from-accumulated-moisture-etc in the fuel tank/line. No replacements of fuel filters helped, as the problem was (stubbornly) well before any fuel filter could remediate.

SOlution was simply: dropping/cleaning the tank, replaced the internal screen. Even replaced all fuel lines ( aluminum, from Aircraft Spruce ) while I was in there.

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'83 300 SD
'05 E320 4Matic
'06 BMW 530 xi
'68 Triumph TR 250 - The only car I ever loved more than the Mercedes; who needs electricity, anyway? - Damn, why did I sell it?!
'59 Jaguar 3.4 'Le Chat Noir' - Damn, why did I sell it?!

It's difficult to make predictions, especially about the future.
- Niels Bohr