The moderate-growth characterization conjecture for pretannakian categories

Let $\mathcal C$ be a pretannakian category, meaning a symmetric tensor category whose objects have finite length. It is incompressible if every tensor functor out of it is an embedding of a tensor subcategory, and it has moderate growth if the lengths of tensor powers of each object are bounded by an exponential function.

Moderate-growth characterization conjecture. The category $\mathcal C$ admits a tensor functor to an incompressible category if and only if it is of moderate growth. Equivalently, every incompressible pretannakian category is of moderate growth.

The forward construction is proved in one direction by the paper's theorem: every moderate-growth pretannakian category has a tensor functor to some incompressible moderate-growth category. The converse characterization remains open.