is the valve spring able to pull back the valve? - if it can then the engine should idle proper unless the camshaft is worn down.

dry black sooty plug is usually caused by excessive fuel in the combustion chamber - try a grease pencil dots on the exhaust manifold runners on cold engine startup and see if there is any variance of its burn time.

quick work can be done with a laser thermometer. your injector to that cylinder seems as the culprit.

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2012 BMW X5 (Beef + Granite suspension model)

1995 E300D - The original humming machine (consumed by Flood 2017)
2000 E320 - The evolution (consumed by flood 2017)