Vanishing of remaining WDVV equations after linear reduction

Prove that, after solving the linear subsystem S_L of WDVV equations for all t^p-free third-order derivatives, the remaining equations S_{NL} vanish identically for arbitrary dimension N and any nondegenerate symmetric metric η, so that the WDVV system can always be written in passive orthonomic form.

Background

From explicit computations, the authors show that in N=4 and N=5 all t^p-free derivatives can be solved linearly and the system is passive in orthonomic form; this leads to commuting Hamiltonian flows.

They conjecture that in general dimensions the remaining (potentially nonlinear) equations vanish once S_L is solved, which would imply universal orthonomic reducibility and compatibility of WDVV.