Non-existence conjecture for endomorphisms satisfying weak Condition dagger

Let $M$ be the closed manifold under consideration, let $Z^1(M)$ denote the space of closed $1$-forms on $M$, and let $\mu$ be an endomorphism of the relevant space of $1$-forms. Condition $\text{weak--}\dagger$ is the technical condition defined in the paper for such an endomorphism. Non-existence conjecture. There is no non-injective, non-trivial endomorphism $\mu$ satisfying Condition $\text{weak--}\dagger$. The conjecture asserts that the natural continuity condition used in constructing pseudo-distances on symplectomorphism groups cannot be realized by any endomorphism with both nontrivial kernel and nonzero action. The paper provides representation-theoretic evidence toward the stronger non-existence phenomenon, including irreducibility of the pullback action of $\operatorname{Diff}_0(M)$ on $Z^1(M)$, but does not establish this conjecture.