the 3570k is a 77W CPU, with no particular notes on the cooling solution that is required.
the 8600k is a 95W CPU, explicitly listed as needing a 130W cooling solution.
So whilst it might manage perfectly well with a 3570k and maybe it gives you a 50% overhead in terms of cooling, that'd still be a 20-30W shortfall for a 8600k.
This is why we've 100's of threads starting with 7th gen 'why is my k so hot' and that simply because it needs a much bigger cooling solution that the tdp would suggest.
It'll probably fit, they've not actually released the 9th gen yet so we don't really know, but they've kept the same configuration since 2nd gen.