The anti-involution conjecture for micro-Kronecker bihamiltonian structures

Let $M$ be a manifold with a micro-Kronecker bihamiltonian structure, and let $m\in M$. For an anti-involution $\alpha$ fixing $m$, its defect is the codimension at the image of $m$ of the fixed-point set induced by $\alpha$ on the local base of the action foliation.

Anti-involution conjecture. There are a neighborhood $U\ni m$ and an anti-involution $\alpha$ of the bihamiltonian structure on $U$ such that

$$\alpha(m)=m$$

and the defect of $\alpha$ at $m$ is $0$.

The conjecture follows formally from the proposed local classification conjecture if that conjecture holds, and is intended to provide canonical symmetries near arbitrary points. The source gives no proof in general.