Bachmann's bi-bracket reduction conjecture

A bi-bracket is an element of the space $\mathcal{BD}$ of bi-brackets, while a mono-bracket is an element of the space $\mathcal{MD}$ of mono-brackets. These spaces are generated over $\mathbb{Q}$ by their respective brackets.

Bachmann's bi-bracket reduction conjecture. Every bi-bracket can be written as a $\mathbb{Q}$-linear combination of mono-brackets; equivalently,

$$\mathcal{BD}=\mathcal{MD},$$

so, in the notation of the source, $BD=MD$.

The conjecture would show that the additional bi-brackets arising from shuffle products introduce no genuinely new elements beyond mono-brackets. It was attributed here to Bachmann; the supplied text gives no evidence that it has been resolved.