Strong Broadhurst–Kreimer and Zagier conjecture for the linearized double shuffle algebra

Let $\mathfrak{ls}$ be the Lie algebra of solutions to the linearized double shuffle relations, let $\mathsf{S}$ denote the space of even period polynomials, and let $\mathsf{e}(\mathsf{S})$ be the subspace generated by the exceptional elements. Write $H_i(\mathfrak{ls};\mathbb Q)$ for Lie algebra homology. Strong Broadhurst–Kreimer and Zagier conjecture.

$$H_1(\mathfrak{ls};\mathbb Q)\cong\mathfrak{ls}_1\oplus\mathsf{e}(\mathsf{S}),$$ $$H_2(\mathfrak{ls};\mathbb Q)\cong\mathsf{S},$$ $$H_i(\mathfrak{ls};\mathbb Q)=0\qquad(i\geq3).$$

This is presented as the strongest Lie-algebraic formulation of the Broadhurst–Kreimer and Zagier conjectures, with substantial but inconclusive numerical evidence. It is intended to imply nearly all the remaining open problems discussed in the paper.