The surjective conjecture for the quasi-uneven quotient

Let $\mathfrak{d}\mathfrak{g}_1$ be the depth-one part of the depth-graded motivic Lie algebra, let $\mathcal{U}\mathfrak{g}$ be its enveloping algebra, and let $\gamma$ be the natural map from $n$ copies of $\mathfrak{d}\mathfrak{g}_1$ to $gr^n_{\mathfrak{D}}\mathcal{U}\mathfrak{g}$. Let $\widetilde{\gamma}$ be the induced map to the quotient of $gr^n_{\mathfrak{D}}\mathcal{U}\mathfrak{g}$ by quasi-uneven elements. Surjective conjecture. For $n\geq3$, the map $\widetilde{\gamma}$ is surjective. This conjecture concerns generation of the quasi-uneven quotient by depth-one motivic Lie elements. The paper does not report a resolution.