Existence of a 1-adjoint singular equivalence that is not 2-adjoint

Construct a singular equivalence of 1-adjoint type between finite-dimensional algebras that is not of 2-adjoint type, or determine that no such singular equivalence exists.

Background

A singular equivalence of 2-adjoint type is a standard singular equivalence whose defining derived tensor functor has a canonical adjoint on the singularity categories, together with the required biperfectness properties. Every singular equivalence of 2-adjoint type is therefore a singular equivalence of 1-adjoint type, but the converse is not established.

The paper provides examples of singular equivalences of 2-adjoint type that are not singular equivalences of Morita type with level, but it does not resolve whether the hierarchy is strict already between the first two levels. The question asks for an example separating 1-adjoint type from 2-adjoint type, or for a proof that such an example cannot exist.

References

Is there a singular equivalence of 1-adjoint type but not 2-adjoint type?

— 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 the question on standard singular equivalences and before the subsection “Constructions”