Complete anisotropy conjecture for the small quantum group of $\mathfrak{sl}_2$

Assume that $\Bbbk$ has characteristic $0$. Let $u_q(\mathfrak{sl}_2)$ be the small quantum group for $q$ an odd root of unity, and let

$$\mathcal{C}:=u_q(\mathfrak{sl}_2)\text{-}\mathsf{Mod}.$$

A non-semisimple modular tensor category is completely anisotropic when its only rigid Frobenius algebra is the unit object.

Complete anisotropy conjecture. The category $\mathcal{C}$ is completely anisotropic: the only rigid Frobenius algebra in $\mathcal{C}$ is the unit object.

Complete anisotropy concerns the absence of nontrivial rigid Frobenius algebras and obstructs the construction of new modular tensor categories by local modules. The conjecture is presented as an example related to Witt equivalence and remains open in the supplied source.