Polynomiality conjecture for the second Hamiltonian structure of the principal hierarchy

Let $P_2^{[0]}$ be the second Hamiltonian structure of the principal hierarchy, and let $P_2$ denote its image under the quasi-Miura transformation

$$w^\alpha=v^\alpha+\epsilon^2\eta^{\alpha\beta}\partial_{1,0}\partial_{\beta,0}\Delta F.$$

Here a Hamiltonian structure has differential polynomial coefficients if its coefficients are differential polynomials in the jet variables. Polynomiality conjecture. The quasi-Miura transformation transforms $P_2^{[0]}$ into a Hamiltonian structure $P_2$ with differential polynomial coefficients. This conjecture concerns the compatibility of the topological deformation with the second Hamiltonian structure; polynomiality is known for the first Hamiltonian structure and for the relevant topological deformation data, while the asserted polynomiality of $P_2$ remains unresolved in the source.