Strongness of the internal hom in closed strong tensor extriangulated categories

Determine whether, in every closed and strong tensor extriangulated category (A, E, s, ⊗, 1) for which the internal hom bifunctor hom(-,-): op(A) × A → A is biextriangulated, the bifunctor hom(-,-) is necessarily strong; equivalently, ascertain whether the natural transformations associated to hom(-,-) satisfy the graded sign-commutativity compatibility across the two variables with respect to cup products on higher extension groups, as required for strong biextriangulated functors.

Background

A tensor extriangulated category is called closed when, for each object X, the functor - ⊗ X admits an extriangulated right adjoint hom(X,-), and the bifunctor hom(-,-) is biextriangulated. The paper introduces a strengthened notion of compatibility for bifunctors, called strong biextriangulated, which encodes a graded sign-commutativity condition for higher extension classes coming from each variable.

While the tensor product in a strong tensor extriangulated category satisfies this stronger compatibility, it remains unsettled whether the internal hom, assumed merely biextriangulated by definition of ‘closed’, automatically inherits the strong property. Resolving this would clarify the behavior of internal homs relative to higher extensions and cup products in the extriangulated setting.