The homological Broadhurst–Kreimer conjecture for the depth graded motivic Lie algebra

Let ${\mathfrak{d}}{\mathfrak{g}}$ be the depth graded motivic Lie algebra, and let $\mathrm{S}$ be the space of cusp forms for $\mathrm{SL}_2(\mathbb{Z})$. Homological Broadhurst–Kreimer conjecture. The homology of ${\mathfrak{d}}{\mathfrak{g}}$ satisfies

$$H_1({\mathfrak{d}}{\mathfrak{g}})\cong\bigoplus_{k\geq 1}\mathbb{Q}\sigma_{2k+1}\oplus\mathrm{S},\qquad H_2({\mathfrak{d}}{\mathfrak{g}})\cong\mathrm{S},$$

and $H_i({\mathfrak{d}}{\mathfrak{g}})=0$ for $i\geq 3$. This conjecturally encodes the structure of the depth graded Lie algebra, including the additional generators associated with cusp-form relations.