Free 2-rig conjecture for the matrix monoid representation 2-rig

Let $k$ be a field and $N$ a positive integer. Let $[?]$ denote the 2-rig of representations of the matrix monoid $[?]$. Free 2-rig conjecture. The 2-rig $[?]$ is the free 2-rig on an object of bosonic subdimension $N$, namely the tautologous representation of $[?]$ on $k^N$. This is one of the conjectures proposed to clarify the structure of 2-rigs; the source gives no resolution.