Steinberg tensor product conjecture for affine group schemes in the Verlinde category

Let $G$ be an affine group scheme with reductive even subgroup $G_0$. Call a representation of $G$ one-irreducible if its restriction to $G_{(1)}$ is irreducible. A representation inflated from $G_0$ is obtained via the Frobenius inflation functor.

Steinberg tensor product conjecture. Any irreducible representation of $G$ is uniquely a tensor product of a one-irreducible representation and one inflated from $G_0$.

This conjecture generalizes the theorem proved in the paper for general linear group schemes and is motivated by expected analogues for orthogonal, symplectic, queer, and periplectic groups in the Verlinde category. The paper notes that these broader representation theories have not yet been classified.