Flatness of factors of a flat product of bihamiltonian structures

Let $M_{1}$ and $M_{2}$ be manifolds equipped with bihamiltonian structures, and equip $M_{1}\times M_{2}$ with their product bihamiltonian structure. Product-flatness conjecture. If $M_{1}\times M_{2}$ is flat, then both $M_{1}$ and $M_{2}$ are flat. The source presents this as an interesting question rather than a resolved result; it would characterize flatness factorwise for products of bihamiltonian structures.