Weak involutivity conjecture for linear gauge invariant PDE systems

Let a system of linear gauge invariant PDEs be given in the form of system~. A system is doubly weakly involutive if it has the corresponding weak involutivity property for both the system and its conjugate, and two systems are weakly equivalent when they are related by the equivalence relation used for system~. Weak involutivity conjecture. Each system of the form~ is weakly equivalent to a doubly weakly involutive system. If true, the degree of freedom of every such system would coincide with that of its conjugate, extending the preceding corollary from systems known to be weakly equivalent to a doubly weakly involutive one to all systems of the form~.