Universal-enveloping-algebra conjecture for the balanced multiple q-zeta Lie algebra

Let $\mathfrak{bm}_0$ be the space of all $\Phi\in\mathbb{Q}\langle\mathcal{B}\rangle$ satisfying $(\Phi\mid b_0)=0$, $(\Phi\mid b_k)=0$ for $k=2,4,6$, primitiveness for the balanced quasi-shuffle coproduct, and $\tau(\Pi_0(\Phi))=\Pi_0(\Phi)$. Let $\mathcal{Z}_q$ be the algebra of balanced multiple $q$-zeta values and let $\widetilde{M}(\operatorname{Sl}_2(\mathbb{Z}))$ denote the algebra of quasi-modular forms. Universal-enveloping-algebra conjecture. The space $\mathfrak{bm}_0$ is a Lie algebra and there is an algebra isomorphism

$$\mathcal{U}(\mathfrak{bm}_0)^\vee\simeq\mathcal{Z}_q/\widetilde{M}(\operatorname{Sl}_2(\mathbb{Z})).$$

This would describe the quotient of balanced multiple $q$-zeta values by quasi-modular forms through the graded dual of a universal enveloping algebra; the source does not state a resolution.