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

Let $k$ be a field of characteristic zero. A 2-rig is the representation-theoretic structure in which the stated universal property is considered; let $x$ be an object, let $n$ be its dimension, let $\Lambda^n(x)$ denote its $n$th exterior power, and let an inverse with respect to the tensor product mean a tensor-invertible object. The general linear representation 2-rig conjecture. The 2-rig $\mathsf{Rep}(\mathrm{GL}(n,k))$ is the free 2-rig on an object of dimension $n$, that is, an object $x$ for which $\Lambda^n(x)$ has an inverse with respect to the tensor product.

This was left as a conjecture in the authors' previous paper and would require techniques beyond those developed here, since $\mathsf{Rep}(\mathrm{GL}(n,k))$ is not a mere quotient of $\overline{k\mathsf{S}}$.