Bachmann's polynomial reformulation of the bracket conjecture

For $r\geq 1$, let $\mathcal{P}_r(N)$ be the set of tuples $(\mathbf{m},\mathbf{n})$ occurring in the coefficient formula for bi-brackets, and let $P\in\mathbb{Q}[X_1,\dots,X_r,Y_1,\dots,Y_r]$. Set

$$M:=r+\sum_{i=1}^r\deg_{Y_i}(P).$$

Refinement of Bachmann's conjecture. There exist polynomials $Q_j\in\mathbb{Q}[X_1,\dots,X_j]$, for $j\geq 1$, with $Q_j\equiv 0$ for all but finitely many $j$, such that for every $N\geq 1$,

$$\sum_{(\mathbf{m},\mathbf{n})\in\mathcal{P}_r(N)}P(m_1,\dots,m_r,n_1,\dots,n_r)=\sum_{j=1}^{M}\sum_{(\mathbf{m},\mathbf{n})\in\mathcal{P}_j(N)}Q_j(n_1,\dots,n_r).$$

This is presented as a reformulation of the conjecture that brackets and bi-brackets span the same space, translating that algebraic claim into coefficient identities for arbitrary polynomials. Its status is therefore the same unresolved status as Bachmann's spanning conjecture.