In my previous question 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 $\operatorname{holim}_nK(X\times \Delta_n)$)?