Implicitness conjecture for symmetric resonance-based schemes

A resonance-based scheme is a numerical method that incorporates dominant oscillatory interactions into its discretisation while approximating the remaining terms. A scheme is symmetric when its time-step map satisfies the usual time-reversibility condition

$$\widetilde{\Phi}_{\tau}=\widetilde{\Phi}_{-\tau}^{-1}.$$

Implicitness conjecture. Every symmetric resonance-based scheme is necessarily implicit.

The conjecture is motivated by the analogous result that symmetric Runge–Kutta methods are necessarily implicit. The paper proposes a symmetric and symplectic resonance-based scheme for the KdV equation that is implicit; whether the same necessity holds for resonance-based schemes in general is left open.