Standardization of singular equivalences

Determine whether every singular equivalence between finite-dimensional algebras is standard, meaning induced up to natural isomorphism by a derived tensor product functor and therefore of 1-adjoint type.

Background

The paper introduces singular equivalences of n-adjoint type as standard singular equivalences equipped with specified chains of adjoint tensor functors whose associated bimodule complexes satisfy biperfectness conditions. A singular equivalence of 1-adjoint type is simply a singular equivalence induced by a derived tensor product functor, without requiring additional adjoints.

The authors explain that singular equivalences are generally difficult to characterize and that their framework encompasses known examples of more refined singular equivalences, including singular equivalences of Morita type with level. The unresolved issue is whether every singular equivalence can be represented in this standard tensor-product form, up to natural isomorphism.

References

Is every singular equivalence standard, i.e., of 1-adjoint type, up to natural isomorphisms?

— Singular equivalences of $n$-adjoint type and standard eventually homological isomorphisms  (2609.03365 - Han et al., 3 Sep 2026) in Section 4, subsection “Comparison with singular equivalences of Morita type with level,” immediately after Example 4.?, before the subsection “Constructions”