Higher dimensional de Rham Theorem

Let $X$ be a smooth manifold, let $x_0\in X$, and let $B_s^n(X)_{x_0}^{hom}$ denote the space of homotopy-invariant higher-dimensional iterated integrals of length at most $s$ on based $n$-loops. Let $\pi_n=\pi_n(X,x_0)$, and let $\pi_n^{<n}$ be the subgroup generated by the images of all maps

$$\alpha_*:\pi_n(S^k)\to\pi_n(X),$$

where $\alpha$ ranges over $\pi_k(X,x_0)$ and $1\leq k<n$. Let $J$ be the augmentation ideal of the group ring $\mathbb Z[\pi_n/\pi_n^{<n}]$. Higher dimensional de Rham Theorem. Iterated integration gives an isomorphism

$$B_s^n(X)_{x_0}^{hom}\stackrel{\cong}{\longrightarrow} \operatorname{Hom}_{\mathbb Z}\left(\mathbb Z[\pi_n/\pi_n^{<n}]/J^{s+1},\mathbb R\right).$$

This conjecture extends Chen's de Rham theorem from one-dimensional loops to higher-dimensional membranes. The paper introduces the higher-dimensional iterated integrals and establishes their basic properties, but the proposed isomorphism is not proved in the supplied text.