The 2-Tillmann conjecture on dualizable objects in 2Kar

An object of the symmetric monoidal $3$-category $\mathbf{2Kar}$ is called $1$-dualizable when it admits a dual in the sense of higher category theory. A finite semisimple $2$-category is a finite semisimple object of $\mathbf{2Kar}$.

2-Tillmann conjecture. An object of the symmetric monoidal $3$-category $\mathbf{2Kar}$ is $1$-dualizable if and only if it is a finite semisimple $2$-category.

This conjecture is a categorification of Tillmann's result and would imply the isofullness needed to identify the semisimple Morita theory with its univalification inside $\mathbf{Mod}(\mathbf{2Kar})$. Its resolution status is not specified in the source.