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Identify the homotopy ind-dg-completion in the non-strongly pretriangulated case

Determine whether, for a pretriangulated dg-category A that is not strongly pretriangulated, the quasi-essential image of the dg-functor L_A^Q: IndOb^{dg}(A) → dgm(A) coincides, up to quasi-equivalence, with the h-projective dg-modules hproj(A). Equivalently, construct a functorial method to handle the additional homotopies needed to totalize perfect dg-modules in this general pretriangulated setting so that the identification Ind^{dg,Q}(A) ≃ hproj(A) holds.

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Background

In Theorem 1.4 (thm - homotopy dg completion closed), the authors show that for an essentially small strongly pretriangulated dg-category A, the homotopy ind-dg-completion Ind{dg,Q}(A) embeds into hproj(A) and this embedding is a quasi-equivalence. The proof uses the strong pretriangulatedness of A to make the totalization of perfect dg-modules Z0-functorial.

When A is only pretriangulated (not strongly pretriangulated), extra homotopies arise in the totalization process, and the authors point out that these complications obstruct extending the identification Ind{dg,Q}(A) ≃ hproj(A). Clarifying whether the quasi-essential image of L_AQ equals hproj(A) in this general setting would complete the picture of the homotopy ind-dg-completion beyond the strongly pretriangulated case.

References

If \A is pretriangulated but not strongly pretriangulated, it is unclear whether the quasi-essential image of \Ll_{\A}{Q}(-) is \hproj(\A), so we do not want to name it the homotopy ind-dg-completion of \A.

— Deformations of triangulated categories with t-structures via derived injectives (2411.15359 - Genovese et al., 22 Nov 2024) in Remark “not strongly pretriangulated,” Subsection “An enhancement of the derived category via filtered homotopy dg-colimits or the homotopy ind-dg-completion” (Section 1.3)