Put it in, close the lid. Job done. Chop it up a bit if you want to speed things along, or run over it with a mower
Do you reckon there's any risk of them just growing in there or being dormant?
Not if you're excluding light. It's possible the roots will survive a while and re-sprout when you use the compost, but the longer you leave it the less likely that is.
@hugo7 started
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Put it in, close the lid. Job done. Chop it up a bit if you want to speed things along, or run over it with a mower