The weight-four hyperlogarithm reduction conjecture

Let $E$ be a field, let ${\cal H}_4(E)$ be the weight-four hyperlogarithmic space, and let $\delta_{2,2}:{\cal H}_4(E)\to\Lambda^2{\cal H}_2(E)$ be the component of $\delta_4$ in $\Lambda^2{\cal H}_2(E)$. Weight-four hyperlogarithm reduction conjecture. Every element of ${\cal H}_4(E)$ annihilated by $\delta_{2,2}$ comes from ${\cal P}_4(E)$. This conjecture is presented as implying the preceding reduction conjecture because ${\cal H}_n(E)={\cal P}_n(E)$ for $n\leq3$; it is a specific unresolved weight-four reduction statement.