The hyperlogarithm-to-polylogarithm reduction conjecture

Let ${\cal H}_n(E)$ and ${\cal P}_n(E)$ be the hyperlogarithmic and polylogarithmic spaces, respectively, with coproduct $\delta_n$, and let $\mathbb{R}H_n$ and $\mathbb{R}P_n$ be their real realizations. Hyperlogarithm-to-polylogarithm conjecture. There exists a natural map

$$r_{{\cal H}_n{\cal P}_n}:\bigl(\operatorname{Ker}\delta_n\subset{\cal H}_n(E)\bigr)\to\bigl(\operatorname{Ker}\delta_n\subset{\cal P}_n(E)\bigr)$$

that satisfies $\mathbb{R}P_n\circ r_{{\cal H}_n{\cal P}_n}=\mathbb{R}H_n$ when $E=\mathbb{C}$. Such a map would reduce the hyperlogarithmic realization to the polylogarithmic one and hence contribute to the proposed reduction of Zagier's conjecture; no general construction is supplied.