The free 2-rig conjecture for representations of the special linear group

Let $k$ be a field of characteristic zero, let $I$ be the unit for the tensor product, and let $x$ be an object of dimension $n$ with $n$th exterior power $\Lambda^n(x)$. The special linear representation 2-rig conjecture. The 2-rig $\mathsf{Rep}(\mathrm{SL}(n,k))$ is the free 2-rig on an object $x$ equipped with an isomorphism

$$\Lambda^n(x) \cong I.$$

The source presents this as one of the conjectures expected to hold for every field of characteristic zero; it is part of the proposed universal characterization of representation 2-rigs of classical groups.