Lagrangianity conjecture for non-Abelian Yang–Mills boundary data

Let $M$ be a manifold with boundary, let $F_M$ be the space of Yang–Mills fields on $M$, and let $F_{\partial M}$ be the space of boundary fields. Let $EL_M$ be the space of solutions to the Euler–Lagrange equations, and let $\pi:F_M\to F_{\partial M}$ be restriction to the boundary. Define

$$L_M=\pi(EL_M).$$

Lagrangianity conjecture. The submanifold $L_M$ is Lagrangian for non-Abelian Yang–Mills theory.

The corresponding statement is established in the Abelian Maxwell case, where the space of boundary values of solutions is shown to be a Lagrangian submanifold. The non-Abelian extension remains unresolved in the supplied source.