Strong Broadhurst–Kreimer–Zagier conjecture for linearised double shuffle Lie algebras

Let $\mathfrak{ls}$ be the Lie algebra of solutions to the linearised double shuffle equations, let $\mathfrak{ls}_1$ be its depth-one part, and let $\mathsf{P}^{e}$ be the space of even period polynomials. Let $\mathbf{e}(\mathsf{P}^{e})$ denote the corresponding space of exceptional generators. For the homology groups $H_i(\mathfrak{ls},\mathbb Q)$, one has:

Strong Broadhurst–Kreimer–Zagier conjecture.

$$H_1(\mathfrak{ls},\mathbb Q)congmathfrak{ls}_1oplusmathbf{e}(\mathsf{P}^{e}),$$ $$H_2(\mathfrak{ls},\mathbb Q)congmathsf{P}^{e},$$ $$H_i(\mathfrak{ls},\mathbb Q)=0quadtext{for all}\quad igeq 3.$$

This is described as the strongest conjecture in the quantitative Broadhurst–Kreimer–Zagier discussion. The source gives no resolution.