Hi
I remove the jaws and spray the scroll while rotating it through the slots with CRC, RP7 or whatever brand of penetrating oil I have to hand. Then while continuing to rotate the scroll I use a wet and dry shop vac with a flat nozzle that fits in the jaw slots to suck out the oil and muck that it has loosened, I usually repeat the process a couple of times. the process works well. the scroll will be clean.
Penetrating Oil is not a good lubricant, (Lightly) oil the Scroll before reassembly with a light oil. not too much or it will attract swarf.
Before reinserting the jaws check the scroll for any swarf that has semi welded itself to the scroll. (And the jaws too)
Oh if you do disassemble a chuck if it is in good condition the scroll is an extremely good fit in the body. It must be removed without tilting or it will get stuck. If it does you will need to tap it with a brass drift through the slots to release it. You may damage the fit in the process if you raise any burrs. Take great care.
Regards
John