Monotonicity of the MZ operation under tensor subcategories

Determine whether for every inclusion of pre-Tannakian categories C ⊂ B, the MZ operation is monotone, i.e., establish whether MZ(C) is necessarily contained in MZ(B); in particular, when C is inert, decide whether MZ(B) must contain C (for example, when C is the Delannoy category).

Background

The paper introduces inert categories (those with MZ(C)=C) and discusses whether objects can be ‘engaged’ to braid nontrivially by embedding into a larger tensor category and taking the Drinfeld center. This leads to a natural structural question about monotonicity of MZ along inclusions.

The authors formulate this as a question about whether MZ respects tensor subcategory inclusions, with particular emphasis on inert subcategories such as the Delannoy category.