Injective free-Lie-algebra map from canonical forms to graph cohomology

Let $\Omega^{\bullet}_{\mathrm{can}}$ be the graded space of canonical differential forms, let $\mathbb{L}(\Omega^{\bullet}_{\mathrm{can}})$ be the free Lie algebra on it, and let $H^n(\mathcal{GC}_2)$ denote the degree-$n$ cohomology of the graph complex. Let $\omega^{4k+1}$, for $k\geq 1$, denote the primitive canonical forms. Graph-cohomology conjecture. There is a non-canonical injective map of graded Lie algebras

$$\mathbb{L}(\Omega^{\bullet}_{\mathrm{can}})\longrightarrow\bigoplus_{n\in\mathbb Z}H^n(\mathcal{GC}_2)$$

whose restriction to the Lie subalgebra generated by the primitive elements maps to the Lie subalgebra of degree-zero cohomology:

$$\mathbb{L}\left(\bigoplus_{k\geq 1}\omega^{4k+1}\,\mathbb Q\right)\longrightarrow H^0(\mathcal{GC}_2).$$

All other elements map to higher-degree cohomology $\bigoplus_{n>0}H^n(\mathcal{GC}_2)$. The grading on the source is by differential-form degree, while the grading on the target is by edge number alone; the degree and loop-number gradings on the target are to be forgotten. This conjecture proposes a non-canonical Lie-algebraic relationship between canonical differential forms and graph cohomology, motivated by graph-integral calculations and their motivic-period analogues; the source does not report a proof or disproof.