Fundamental decoupling identity for generalized color orderings

Let $CO_{k,n}$ denote the set of generalized color orderings of type $(k,n)$. A triple $\mathcal{T}=(\Sigma_0,\Sigma_1,\Sigma_2)$, with

$$\Sigma_0\in CO_{k-1,n-2},\qquad \Sigma_1\in CO_{k-1,n-1},\qquad \Sigma_2\in CO_{k,n-1},$$

is fundamental when $\pi_i(\Sigma_1)=\pi_{(j)}(\Sigma_2)=\Sigma_0$. A generalized color ordering $\Sigma\in CO_{k,n}$ is a double $\mathcal{T}$-extension of $\Sigma_0$ when $\pi_{(i)}(\Sigma)=\Sigma_1$ and $\pi_j(\Sigma)=\Sigma_2$ for some $i\ne j$.

Fundamental decoupling identity. Fix a fundamental triple $\mathcal{T}=(\Sigma_0,\Sigma_1,\Sigma_2)$. As $\Sigma\in CO(k,n)$ ranges over all double $\mathcal{T}$-extensions $\Sigma$, the CEGM integrands $\mathcal{I}(\Sigma)$ satisfy a unique linear identity whose coefficients are all $\pm1$.

This identity is proposed as a refinement compatible with the duality between the configuration spaces $X(k,n)$ and $X(n-k,n)$, and is called a fundamental decoupling identity. The supplied text does not establish whether the asserted uniqueness and coefficient condition have been proved, so the status remains open.