Uniqueness of generalized tensor product categorifications

Establish whether an analogous uniqueness theorem holds for generalized tensor product categorifications of mixed highest- and lowest-weight modules under the additional assumptions on the projective object \(\bar P(b_0)\).

Background

The paper defines generalized tensor product categorifications that allow highest- and lowest-weight modules to be mixed. It explains that the usual uniqueness proof for tensor product categorifications relies on the subcategories generated by the distinguished projective objects being equivalent to module categories over Frobenius algebras. In the mixed setting, the relevant projective and injective objects do not generally coincide, so that argument does not apply. The unresolved question is whether uniqueness can nevertheless be proved after imposing additional assumptions on the distinguished projective object $\bar P(b_0)$.