Conjecture on Frobenius-structure eigenvalues and Hamiltonian spectral numbers

Let $H$ be a non-degenerate Hamiltonian on $M$, let $\tilde U_{\xi}$ be the associated space, and let $\star:T U\rightarrow \operatorname{End}(E)$ be the Frobenius structure over $U$ with Euler vector field $X_E$. For $a\in H_*(\tilde U_{\xi},\mathbb{Z})$, write $\rho(H,a)$ for the corresponding spectral number of $H$. The ‘eigenvalues’ of $\nabla X_E$ over $\tilde U_{\xi}$, namely the spectrum of the Frobenius structure and the spectral numbers of the associated variation of Hodge structures, Frobenius-spectrum conjecture. coincide generically, after eventual affine scaling, with the spectral numbers $\rho(H,a)$ as $a$ ranges over all elements of $H_*(\tilde U_{\xi},\mathbb{Z})$. This conjecture proposes a correspondence between spectral invariants from Hamiltonian dynamics and the spectrum arising from the associated Frobenius and Hodge-theoretic structures; the supplied text does not state whether it is known or open.