Because of the matte finish on those B pillars, I'd think twice before hitting them with a machine and compound/polish.
Personally I'd be very inclined to try chemical or non-abrasive means before resorting to abrasives, lest you change the appearance of the finish permanently.
You could try clay or synthetic clay mitts, some APC (with a water rinse) on microfiber, a non-abrasive pre-wax "cleaner" (something like the old Zymol HD Cleanse or P21S paintwork cleanser come to mind) and such.
I haven't faced this particular challenge, but I had decided to machine polish/compound the matte B pillars on my E39 to remove oxidation and ended up with "semi-gloss". In my situation, I expected this, but that may not be what you're after.
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