I believe the hot idle shake is related to fuel lubricity. When diesel gets hot, the lubricity goes down so that would explain why it only happens when hot. But why would less lubricating fuel cause the engine to shake? I suppose another factor is fuel energy.

In my case I've noticed that when running WVO, the shakes go completely away. B100 , hardly any shakes, B20 mild only after long interstate drive. ULSD -- shakes pretty good after car gets hot. So try some b100 and see if your hot shakes go away or some fuel lubricity additive and see if that helps.

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What Would Rudolph Do?
1975 300D, 1975 240D, 1985 300SD, 1997 300D, 2005 E320 , 2006 Toyota Prius