Brauer-loop groundstate and upper-upper scheme degree correspondence conjecture

Let $\pi\in S_n$ be a permutation, and let $\overline{E}_\pi$ be the corresponding subset of the upper-upper scheme for pairs of $n\times n$ matrices. For odd size, let $\pi$ be a partial permutation matrix of rank $n$, and let $\overline{E}_\pi$ be the corresponding subset for an $n\times(n+1)$ matrix and an $(n+1)\times n$ matrix.

Brauer-loop degree correspondence conjecture. The groundstate element of the Brauer Hamiltonian for $L=2n$ corresponding to $\pi\in S_n$ equals the degree of $\overline{E}_\pi$. For $L=2n+1$, the groundstate element corresponding to the rank-$n$ partial permutation matrix $\pi$ equals the degree of the corresponding subset $\overline{E}_\pi$ of the upper-upper scheme.

The conjecture is motivated by explicit agreement for the cases computed in the source, including permutations in $S_3$ and $S_4$, and analogous calculations for odd system sizes. Its general validity remains open in the source.