For a smooth manifold $X$, let $B_s(X)$ denote the space of iterated integrals of length at most $s$. Here we consider iterated integrals as functions on the path space $PX$. Fix a base point $x_0$ and consider the set of paths $PX_{x_0}$ of all paths emanating at $x_0$. Denote the iterated integrals on this space by $B_s(X)_{x_0}$. Let $LX_{x_0}$ be the loop space, i.e., the set of all paths emanating and ending at $x_0$. Let $B_s(X)_{x_0,x_0}$ be the iterated integrals on this space. Clearly, the restriction map $B_s(X)_{x_0}\to B_s(X)_{x_0,x_0}$ is surjective. Now let $B_s(X)_{x_0}^{hom}$ be the subset of iterated integrals which are invariant under homotopies with fixed endpoints. Likewise define the space $B_s(X)_{x_0,x_0}^{hom}$.

The QUESTION is, whether the restriction map $$ B_s(X)_{x_0}^{hom}\to B_s(X)_{x_0,x_0}^{hom} $$ again is surjective. In the cases $s=0,1$ this is easily established. It sounds like a question you might find in Chen's papers, but I didn't. But then again, I am not a specialist in this area and no good reader, so I might have overlooked something helpful.