Conjecture on free relations for symplectic matrix invariants in characteristic two

Let $p$ be the characteristic of the ground field, let $R^{S\!p(n)}$ be the invariant ring for $n\times n$ generic matrices under the symplectic group, and let $K_{\infty}$ denote the ideal of free relations, namely the intersection of the relation ideals $K_i$ over all positive matrix sizes. Let $\mathcal{N}$ be the algebra of formal generators, and write $a\stackrel{c}{\sim}a^T$ when $a$ is cyclically equivalent to its transpose. Free-relations conjecture. If $p=2$, then the ideal $K_{\infty}$ of free relations for $R^{S\!p(n)}$ is generated by the elements $\sigma_t(a)$ for $a\in\mathcal{N}$ satisfying $a\stackrel{c}{\sim}a^T$ and odd $t>0$. The theorem preceding this conjecture establishes that free relations vanish for orthogonal and symplectic invariants when $p\neq2$; the characteristic-two case is presented as the remaining exceptional case, with the proposed generators described in the conjecture.