Virasoro symmetry conjecture for deformations of the Principal Hierarchy

Let the Principal Hierarchy be associated to a semisimple Frobenius manifold, and consider a deformation of its bihamiltonian structure with constant central invariants. Denote the Virasoro symmetries of the hierarchy by $\frac{\partial}{\partial s_m}$ and the corresponding deformed integrable hierarchy by the deformation determined by the bihamiltonian structure. Virasoro symmetry conjecture. For any such deformation, the Virasoro symmetries $\frac{\partial}{\partial s_m}$ can be deformed to be symmetries of the deformed integrable hierarchy. The conjecture is verified for $m=-1$; the source states that a proof for the general case is planned in the second paper of the series.