Resolving this question would establish a universal finite-moment principle for chaos convergence across the full landscape of discrete probability, mirroring the completeness that the fourth-moment theorem brought to Gaussian analysis. It would create practical statistical tests for Poisson-type behavior in systems built from non-standard noise, relevant to random graph theory, combinatorial stochastic processes, and high-dimensional statistics. It would also clarify the deeper reason why four moments suffice at all, potentially revealing a unifying algebraic obstruction tied to the genus of interaction diagrams rather than to the specific choice of noise.