The free 2-rig conjecture for representations of the symplectic group

Let $k$ be a 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 symplectic representation 2-rig conjecture. The 2-rig $\mathsf{Rep}(\mathrm{Sp}(n,k))$ is the free 2-rig on such an object $x$ whose counit is antisymmetric:

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

The source expects this universal characterization to hold for every field of characteristic zero, as part of a program for characterizing representation 2-rigs of classical groups.