The central-invariant criterion for linearizable Virasoro symmetries

Let a semisimple Frobenius manifold have an associated Principal Hierarchy, and let $(P_0,P_1)$ be a tau-symmetric bihamiltonian deformation of that hierarchy. The deformation has central invariants given by the central invariants of its corresponding deformed bihamiltonian structure, and its Virasoro symmetries are called linearizable when a Miura type transformation can make all differential-polynomial terms in the transformed symmetries vanish. Linearization conjecture. The Virasoro symmetries are linearizable if and only if all central invariants of the corresponding deformed bihamiltonian structure equal $\frac{1}{24}$. The criterion is motivated by the one-dimensional example, where linearization occurs for central invariant $\frac{1}{24}$, and by an example in which the Virasoro symmetries cannot be linearized because the required transformation does not exist. The general equivalence remains open in the supplied source.