Web-rigidity conjecture for homogeneous bihamiltonian structures

Let $M$ and $M'$ carry homogeneous bihamiltonian structures

$$\left(M,\left\{\right\}_{1},\left\{\right\}_{2}\right),\qquad \left(M',\left\{\right\}'_{1},\left\{\right\}'_{2}\right).$$

For small open subsets $U\subset M$ and $U'\subset M'$, let ${\cal B}_{U}$ and ${\cal B}_{U'}$ be the corresponding webs. Web-rigidity conjecture. If ${\cal B}_{U}$ and ${\cal B}_{U'}$ are locally isomorphic, then the bihamiltonian structures on $M$ and $M'$ are locally isomorphic; in particular, their types coincide. The conjecture proposes that the associated webs determine the local bihamiltonian geometry of homogeneous structures; the source gives no resolution.