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Hmm.. Thank you very much. So there is at least a 'high chance' that it works in 'good situations'. That is indeed good to know. Anyways, if anyone has an actual proof, it would be much appreciated if you could give a citation. Thanks again.
Thanks for the answer. But then, what about doing the descant data twice, or probably 3 times, i.e. defined $A''(U):=2$-$lim_{\mathfrak{U}}(Des(\mathfrak{U},A'))$ and then again $A'''$. Would that work?