Goncharov's Chern character conjecture for the formal Hopf algebra

Let $F$ be an infinite field, let $1\leq i\leq n$, and let $\mathcal{H}^{\mathrm{f}}(F)$ denote the formal Hopf algebra with weight-$n$ cohomology $H^i(\mathcal{H}^{\mathrm{f}}(F),\mathbb{Q})_n$. Goncharov's Chern character conjecture. There exist isomorphisms

$$\operatorname{ch}_{n,i}\colon \operatorname{gr}_{\gamma}^{n}K_{2n-i}(F)_{\mathbb{Q}}\longrightarrow H^i(\mathcal{H}^{\mathrm{f}}(F),\mathbb{Q})_n.$$

This predicts that the formal Hopf algebra has the expected graded $K$-theoretic cohomology; the source does not state a resolution status.