Existence of a first-order Hamiltonian operator for all first-order WDVV systems

Show that every first-order system of conservation laws obtained from the WDVV equations by choosing a distinguished independent variable t^p admits a Ferapontov-type first-order Hamiltonian operator A1^{ij} = g^{ij}D_x + Γ^{ij}_k u^k_x + c^{αβ} w^i_{α q} u^q_x D_x^{-1} w^j_{β p} u^p_x that is compatible with the third-order Dubrovin–Novikov operator constructed in this work.

Background

The authors construct, for arbitrary dimension, third-order Hamiltonian operators for the first-order WDVV systems derived via linear line congruences, and prove their uniqueness (up to scaling) via the Metric Theorem. In dimensions N=4 and N=5 with η=η^{(2)} they also verify the existence of a compatible first-order Ferapontov-type operator by direct computation.

They report that a general proof of existence of a first-order Hamiltonian operator for all first-order WDVV systems is lacking; resolving this would establish a full bi-Hamiltonian structure for WDVV in arbitrary dimension.