Rigidity of non-negligible objects of moderate growth in braided categories (via Zoom)

Let k be a field, and let C be a Cauchy complete k-linear braided category with finite dimensional morphism spaces and End(1)=k. We call an indecomposable object X of C non-negligible if there exists Y in C such that 1 is a direct summand of Y⨂X. We prove that every non-negligible object X in C such that dim(End(X^(⊗n))