In my previous question http://mathoverflow.net/questions/74237/the-vanishing-of-non-connective-k-theory-in-negative-degrees I asked when one can be sure that the negative non-connective $K$-groups of a differential graded category vanish. Now, does the situation become simpler if one considers the homotopy invariant $K$-theory (so, one replaces the spectrum $\mathcal{K}(X)$ by $\operatorname{holim}_nK(X\times \Delta_n)$)?
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