Notation: Let $X$ denote a smooth manifold (without boundary) and define $LX = C^{\infty}(S^1, X)$ to be the loop space on $X$.
In the context of loop space homology and the supersymmetric path integral on $LX$ one considers certain differential forms on $LX$ which are given by Chen's iterated integrals. On the one hand, one may consider $LX$ as a Fréchet manifold modeled on $C^{\infty}(S^1, \mathbb{R}^{\dim(X)})$ and with that comes a notion (as in Chapter 33 of Kriegl/Michor) of a differential $k$-form as an element of $C^{\infty}(S^1, L^k_{alt}(T(LX), \mathbb{R}))$. On the other hand, it is helpful to relax the conditions on $X$. Instead of a smooth manifold it suffices to work with generalized smooth spaces whose smooth structure is defined in terms of plots.
For a diffeological space $Y$ a differential form $\omega$ is a collection $\omega_\varphi \in \Omega(U)$ of differential forms for each plot $\varphi \colon U \to Y$. If we consider $Y = LX$ as a diffeological space, every differential form $\omega$ on $LX$ in the sense of Fréchet manifolds determines a diffeological form on $LX$ via pullback: $\omega_{\varphi} = \varphi^*\omega$.
Question: Can every diffeological differential form on $LX$ be regarded as a pullback of a single differential form on $LX$ in the sense of Fréchet manifolds?
