Bachmann's conjecture that brackets and bi-brackets span the same space

Let $\mathcal{MD}$ be the algebra of brackets and $\mathcal{BD}$ the $\mathbb{Q}$-algebra of bi-brackets, both viewed as subalgebras of the algebra $\mathcal{Z}_q$ of $q$-multiple zeta values. Write $\mathcal{Z}_q^\circ$ for the bracket subalgebra.

Bachmann's conjecture. We have

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

i.e. $\mathcal{Z}_q^\circ=\mathcal{Z}_q$.

The bi-bracket algebra is known to equal $\mathcal{Z}_q$, so the conjecture asserts that brackets already generate all $q$-multiple zeta values. It was formulated by Bachmann based on comparisons of dimensions in small weights.