The local classification conjecture for micro-Kronecker bihamiltonian structures

Let $M$ be a manifold with a micro-Kronecker bihamiltonian structure. Suppose the action foliation restricts over $U_0\subset M$ to a fibration with base $\cal B_{U_0}$, let $m\in U_0$, and let $b$ be the projection of $m$ to $\cal B$. Let $\cal W$ be the induced Kronecker web on $\cal B_{U_0}$, and let $m'\in\Phi(\cal W)$ be the point on the zero section over $b$.

Local classification conjecture. There is a local isomorphism of bihamiltonian structures between $M$ near $m$ and $\Phi(\cal W)$ near $m'$.

This conjecture is presented as a generalization of an earlier conjecture and would reduce local classification to the induced Kronecker web, or equivalently to action variables while ignoring angle variables. The analytic rank-one case is claimed in the cited literature, and the smooth rank-one case is stated to have been proved; the general conjecture remains open.