Orthonomic reducibility of WDVV equations in all dimensions

Determine whether, for arbitrary dimension N and any nondegenerate symmetric metric η, the WDVV equations reduce entirely to the orthonomic system f_{ijk} = G^{ijk}(f_{plm}) (equation (28)) after solving all t^p-free third-order derivatives in terms of t^p-derivative variables; equivalently, ascertain whether the remaining subsystem S_{NL} vanishes identically on account of S_L.

Background

The paper splits WDVV into a linear subsystem S_L (solvable for t^p-free derivatives) and a remaining subsystem S_{NL}. In dimensions N=4 and N=5 (for arbitrary η), they prove that the full system reduces to the orthonomic form (28) and is passive, which implies commutativity of the derived Hamiltonian systems.

It remains unknown whether this reduction holds universally. Deciding this would settle the general commutativity and compatibility program proposed by the authors.