Deitmar–Horozov conjecture on restriction of closed iterated integrals

Let $X$ be a smooth manifold, let $x_0\in X$, and let $B_s(X)^{\rm hom}$ denote the space of closed iterated integrals of length at most $s$, meaning iterated integrals invariant under homotopies with fixed endpoints. Let $B_s(X)_{x_0}^{\rm hom}$ be its restriction to paths beginning at $x_0$, and let $B_s(X)_{x_0,x_0}^{\rm hom}$ be its restriction to loops based at $x_0$.

Deitmar–Horozov conjecture. The restriction map

$$B_s(X)_{x_0}^{\rm hom}\to B_s(X)_{x_0,x_0}^{\rm hom}$$

\nis surjective.

The conjecture asserts that every closed iterated integral on based loops extends to a closed iterated integral on paths beginning at the base point. Its proposed consequence is that the module of higher-order invariants of smooth functions is generated by free closed iterated integrals; the supplied text gives no resolution status.