The generalized ideq-tilting equivalence conjecture

Let $\mathcal{E}$ be an exact category, and suppose that there exists an integer $n\geq 0$ for which $\mathcal{E}$ has at least one $n$-tilting subcategory. Let $\mathcal{P}^{<\infty}$ denote the objects of finite projective dimension, and let $\operatorname{Thick}_{\Delta}(\mathcal{P}^{<\infty})$ be the thick subcategory they generate. Generalization of the theorem on ideq tilting. The following conditions are equivalent:

1. There exist $n\geq 0$ and an $n$-tilting subcategory that is ideq $n$-tilting.
2. For every $m\geq 0$, every $m$-tilting subcategory is ideq $m$-tilting.
3. There is an idempotent complete additive category $\mathcal{S}$ and a triangle equivalence

$$D^b(\mathcal{E})\longrightarrow D^b(\operatorname{mod}_{\infty}-\mathcal{S})$$

that restricts to a triangle equivalence

$$\operatorname{Thick}_{\Delta}(\mathcal{P}^{<\infty})\longrightarrow K^b(\mathcal{S}).$$

The conjecture proposes an equivalence between the existence of one ideq tilting subcategory, the corresponding property for all tilting subcategories, and a derived-category description by an additive category. Its status is not resolved in the supplied text and is presented as a conjectural consequence of an extension property for arbitrary triangle equivalences.