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Are weak nilpotent extensions always nilpotent extensions?

Determine whether every weak nilpotent extension f: C → D of pre-c-categories (in the sense that f is 0-connective and any sufficiently long composite of morphisms killed by f vanishes) is necessarily a nilpotent extension of pre-c-categories, i.e., whether the associated algebra f^*f_! in Alg(End_Pr^L(Ind(C))) is nilpotent in the sense of Definition 4.16. Equivalently, ascertain whether the converse of the implication “nilpotent extension ⇒ weak nilpotent extension” holds beyond the case of bounded weighted categories.

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Background

The paper defines two notions for morphisms of pre-c-categories: nilpotent extensions via the associated algebra f*f_! in the endofunctor category of Ind(C), and weak nilpotent extensions via vanishing of sufficiently long composites of morphisms annihilated by f.

Proposition 4.23(2) shows that a nilpotent extension is a weak nilpotent extension. The authors can prove the converse when both source and target are bounded weighted categories, but not in general.

References

Proposition~\ref{prop:weak_nilpotence}(2) implies that a nilpotent extension of pre-$c$-categories $f:\cC \to \cD$ is a weak nilpotent extension. We do not know whether the converse holds in general, however, it is true if both $\cC$ and $\cD$ are bounded weighted categories: in this case we have an equivalence $\Ind(\cC) \simeq \Fun_{\Sigma}(\cC{\heartsuit, \mathrm{op}}, \Sp)$, and triviality of the map $I{\otimes m} \to \id_{\Ind(\cC)} \to \pi_0(\id_{\Ind(\cC)})$ can be checked objectwise. In this case, the notion of nilpotent extension here agrees with that of Definition 5.1.1.

— $c$-structures and trace methods beyond connective rings (2509.14774 - Levy et al., 18 Sep 2025) in Remark, Section 4.2 (Nilpotent extensions and connective maps)