Red Pikmin only: Guard Satellite theory
Looks like this is a hard one.
You can get Olimar into the arena, you can get pikmin there, you can even kill BLL. But so far I haven't managed to get it out.
I have yet to get GS close to the gate due to the effects of the slope. If you have the part as close to the gate as possible while touching either wall, the next day it remains behind the gate, inaccessible once more. You can re-enter the arena and attempt to push the part with Olimar, but cannot enter with pikmin due to puffstool never respawning.
I still have theories on this. You can move the part as close to the gate as you like by swarming pikmin near the part and dismissing so they immediately grab the part, minimising sliding. If we can get the part into the corners of the wall and go to sunset, it may be possible.
There's also the myth that rarely BLL leaves his arena (possibly began as a troll on the pikmin wiki). This would be the easiest option, however nobody in the pikmin community has ever replicated this. We would just have to kill bll as his core is over the wall, and presto, part get. I don't have my hopes up for that though.
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