Brauer-loop groundstate integrality, concatenation, and maximality conjecture

Let $\psi_0$ be the groundstate of the Brauer Hamiltonian, whose entries are weights of chord diagrams. For odd system size $L=2n+1$, a partial permutation has rank $n$; for even size $L=2n$, the relevant diagrams correspond to permutations. If $\pi_n$ and $\pi_m$ are permutations, with one possibly a partial permutation, define their concatenation $\pi_{n+m}$ by

$$\pi_{n+m}(i)=\pi_n(i)\quad (1\leq i\leq n),\qquad \pi_{n+m}(n+i)=n+\pi_m(i)\quad (1\leq i\leq m).$$

Brauer-loop groundstate conjecture. (i) For each system size, the overall normalisation can be chosen so that the smallest element of $\psi_0$ is $1$ and all other elements are integers. (ii) With this normalisation, the weight of a chord diagram corresponding to the (partial) permutation $\pi_{n+m}$ equals the product of the weights of the chord diagrams corresponding to $\pi_n$ and $\pi_m$. (iii) Among the weights of chord diagrams corresponding to permutations, the weight associated with the long permutation $w=(n,n-1,\ldots,2,1)$ is largest.

These are numerical conjectures about the arithmetic and multiplicative structure of the Brauer-loop groundstate. The source provides numerical evidence but no resolution of these assertions.