Arthamonov's conjecture on modified double Poisson brackets

Let $\mathbb{K}$ be the ground field and let $A=\mathbb{K}\langle x_1,x_2,x_3\rangle$ be the free associative algebra on generators $x_1,x_2,x_3$. Define two operations on pairs of generators by

$$\begin{aligned} &\{\!\!\{ x_1,x_2 \}\!\!\}^I=-x_2x_1 \otimes 1, \qquad &&\{\!\!\{ x_2,x_1 \}\!\!\}^I=x_1x_2 \otimes 1, \\ &\{\!\!\{ x_2,x_3 \}\!\!\}^I=-x_2\otimes x_3, \qquad &&\{\!\!\{ x_3,x_2 \}\!\!\}^I=x_2 \otimes x_3, \\ &\{\!\!\{ x_3,x_1 \}\!\!\}^I=-1 \otimes x_3x_1, \qquad &&\{\!\!\{ x_1,x_3 \}\!\!\}^I=1\otimes x_1x_3, \end{aligned}$$

and

$$\begin{aligned} &\{\!\!\{ x_1,x_2 \}\!\!\}^{I\!I}=-x_1 \otimes x_2, \quad &&\{\!\!\{ x_2,x_1 \}\!\!\}^{I\!I}=x_1 \otimes x_2, \\ &\{\!\!\{ x_2,x_3 \}\!\!\}^{I\!I}=x_3\otimes x_2, \quad &&\{\!\!\{ x_3,x_2 \}\!\!\}^{I\!I}=-x_3 \otimes x_2, \\ &\{\!\!\{ x_3,x_1 \}\!\!\}^{I\!\!}=x_1 \otimes x_3 - x_3 \otimes x_1. \end{aligned}$$

All remaining omitted terms involving pairs of generators are zero. Arthamonov's conjecture. These two operations define modified double Poisson brackets on $A$. The conjecture proposes two additional examples of modified double Poisson brackets, extending the operations from generators according to the relevant derivation rules. The examples are intended to provide computable noncommutative Poisson structures whose representation-space brackets descend to representation moduli spaces. The source supplies no resolution status, so the conjecture remains open.