Boundary-changing E2 generalized Eilenberg–Watts equivalence

Establish that, for E2-algebras B1, B2, B3, and B4 in an E2-monoidal category, E1-algebras A1 and A2 equipped with compatible B1–B2 and B3–B4 actions, and E2-algebra maps f1:B1→B3 and f2:B2→B4, tensoring with a B3–B4-bi-A1–A2-bimodule induces an equivalence between the category of such bi-bimodules and the category of topological boundary-changing bimodule functors between the corresponding representable module-category bimodules.

Background

The paper constructs an E2 Eilenberg–Watts equivalence for fixed endpoint data, identifying bi-bimodules with bimodule functors between representable module categories. It then enlarges the target to allow nontrivial side-condensation maps f1:B1→B3 and f2:B2→B4, so that the horizontal boundaries of a 2-cell may have different endpoint objects.

The unresolved issue is whether the fixed-endpoint equivalence extends to these boundary-changing cells. The conjecture asserts that a bi-bimodule M induces the corresponding functor by relative tensor product, and that every compatible topological boundary-changing bimodule functor arises in this way.