Classical representation conjecture for null Lagrangians on diffeomorphism groups

Let $M$ be a manifold, let $nl^{0}(M)$ denote the space of null Lagrangians of order zero on $M$, and let $J_{c}(M)$ denote the relevant compactly supported jet space. A classical null Lagrangian is a function represented by the standard classical null-Lagrangian construction. Classical representation conjecture. For any $M$ and any $W\in nl^{0}(M)$, there is a classical null Lagrangian $\tilde{W}$ such that

$$W(\mathbf{F})=\tilde{W}(\mathbf{F})$$

for any $\mathbf{F}\in J_{c}(M)$. The conjecture proposes that every order-zero null Lagrangian on a diffeomorphism group agrees on compactly supported jets with a classical null Lagrangian; the supplied text gives no resolution or further scope of what is known.