Unique Ferapontov Hamiltonian-operator conjecture for first-order WDVV systems

Let $A_1$ be a first-order Hamiltonian operator of Ferapontov type for a first-order WDVV system, compatible with the third-order Hamiltonian operator supplied by the Hamiltonian theorem. Let $\tilde{\eta}=(\eta^{ij})_{i,j\in\{2,\ldots,N\}}$. Unique Ferapontov-operator conjecture. Every first-order WDVV system admits a unique such operator $A_1$; its constants satisfy

$$(c^{\alpha\beta})_{\alpha,\beta=1}^{N-1}=\tilde{\eta}\det\tilde{\eta},$$

and $A_1$ is local if and only if $\tilde{\eta}$ is degenerate. The conjecture is motivated by the bi-Hamiltonian pairs found in all examples treated by the authors; its validity in general is not established in the supplied text.