Resolving this question would provide a comprehensive quantitative framework for approximation errors in CLTs applicable to a much broader class of models used in finance, physics, and communications, where heavy tails and switching dynamics are standard. It would likely require developing new Stein operators adapted to stable distributions, new coupling arguments for regime-switching chains, and possibly new interpolation inequalities between Wasserstein metrics of different orders. Success would also clarify whether Wasserstein-2 is even the right metric in the heavy-tailed regime or whether weaker transport metrics must replace it, shaping future directions in both theoretical probability and its applications.