The free 2-rig conjecture for representations of the 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 a self-dual object of dimension $n$. Let $\epsilon\colon x\otimes x\to I$ be its counit, and let $S_{x,x}$ denote the symmetry isomorphism exchanging the two copies of $x$. The orthogonal representation 2-rig conjecture. The 2-rig $\mathsf{Rep}(\mathrm{O}(n,k))$ is the free 2-rig on such an object $x$ whose counit is symmetric:

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

The algebraic-closedness assumption ensures that all nondegenerate symmetric bilinear forms on an $n$-dimensional vector space over $k$ are isomorphic; the conjecture is another proposed universal characterization for a classical-group representation 2-rig.