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

Let $k$ be an algebraically closed field of characteristic zero, let $I$ be the unit for the tensor product, and let $x$ be an object of dimension $n$ equipped with an isomorphism $\Lambda^n(x)\cong I$. Suppose also that $x$ is self-dual with symmetric counit $\epsilon\colon x\otimes x\to I$. Let $S_{x,x}$ denote the symmetry isomorphism. The special orthogonal representation 2-rig conjecture. The 2-rig $\mathsf{Rep}(\mathrm{SO}(n,k))$ is the free 2-rig on an object $x$ with these properties; in particular, its symmetric counit satisfies

$$\epsilon\circ S_{x,x}=\epsilon.$$

The source places this conjecture after the orthogonal-group statement under the algebraically closed characteristic-zero hypothesis, extending the proposed universal descriptions to another classical group.