Fermionic reformulation of dispersive universal Whitham hierarchies

Let $C$ be any spectral curve, and let the dispersive universal Whitham integrable hierarchy associated with $C$ be the corresponding hierarchy. Fermionic reformulation conjecture. There is a fermionic reformulation of dispersive universal Witham integrable hierarchy associated with any spectral curve. The conjecture extends fermionic descriptions known for special cases of partition functions and is intended to relate dispersive universal Whitham hierarchies to fermionic formalisms; the source mentions partial results but no general proof.