Spectral correspondence between Frobenius eigenvalues and Hamiltonian spectral numbers

Let $H$ be a non-degenerate Hamiltonian on $M$, let $U$ be the base of the associated Frobenius structure on a vector bundle $E$, and let $X_E$ be its Euler vector field. The 'eigenvalues' of $\nabla X_E$ are defined by evaluating the closed part, via Hodge decomposition, of the $\operatorname{End}(E)$-valued one-form associated to $\nabla X_E$ on a set of generators of $H_1(U,\mathbb{Z})$. Let $\xi\in H^1(U,\mathbb{R})$ and let $\widetilde U_\xi$ be the corresponding space used to define the spectral numbers $\rho(H,a)$ for $a\in H_*(\widetilde U_\xi,\mathbb{Z})$. Spectral correspondence conjecture. The 'eigenvalues' of $\nabla X_E$ over $U$, equivalently the spectrum of the Frobenius structure $\star:TU\rightarrow\operatorname{End}(E)$ and the spectral numbers of the variation of Hodge structures associated to $H$, coincide generically, after eventual affine scaling, with the spectral numbers $\rho(H,a)$ of $H$ as $a$ ranges over all elements of $H_*(\widetilde U_\xi,\mathbb{Z})$. This conjecture proposes a correspondence between the spectral invariants of Hamiltonian dynamics and the spectral data of the associated Frobenius and Hodge structures; the paper presents it as a conjectural complement to Theorem 1, and no resolution is supplied in the given text.