In a recent analysis of the exclusive production of $\eta-\pi$ and $\eta’-\pi$ meson systems with a 191 GeV/c pion beam on a proton target at COMPASS, an unexpected enhancement of $\eta’-\pi$ over $\eta-\pi$ was observed for the odd partial waves. These carry exotic quantum numbers therefore cannot be associated with the conventional quark-antiquark states. The collected data covers a wide range of the dimeson invariant masses extending to the region where the multi-particle production process enters the diffractive regime and can be well described by means of the theory of complex angular momentum, namely in terms of the $t$-channel exchanges of Regge trajectories. Assuming the analyticity and the multi-Regge behavior of the scattering amplitude at high energies, the finite energy sum rules for multiple production can be formulated. The sum rules give a consistency condition between a quasi-two-body Regge amplitude and a double Regge amplitude in the case of reactions with three particles in the final state. Such conditions allow to examine the analytical structure of the multi-particle production amplitudes and to shed light on the observed $\eta-\pi$ puzzle.