Poisson commutativity of the conserved quantities for the $B_r$ Q-system

Let $A_1,\dots,A_{2r}$ be coordinates on the phase space of the $B_r$ Q-system, and let $B$ be the signed adjacency matrix of its quiver. Define the Poisson bracket by

$$\{A_i,A_j\}=\Omega_{ij}A_iA_j,$$

where

$$\Omega=(B^T)^{-1}=-B^{-1}.$$

The conserved quantities, namely the Hamiltonians of the $B_r$ Q-system, Poisson-commutativity conjecture. The Hamiltonians Poisson-commute under this Poisson bracket.

This would establish the expected integrability property for the $B_r$ Q-system. The paper reports experimental evidence for the claim, but the statement is presented as a conjecture rather than proved in the supplied text.