The problem is open because Stein's method for Wasserstein-2 bounds relies critically on controlling second moment quantities and on smoothness of test functions, both of which break down under heavy-tailed or stable regimes where variance may be infinite. Regime-switching introduces additional correlation structure that couples across switching times in a way that invalidates standard mixing or cumulant truncation arguments. Stable-like processes can have self-similar scaling limits that are not Gaussian, so even deciding when a CLT in the Wasserstein-2 sense applies versus when a stable limit theorem takes over requires new criteria. The interplay between the periodicity of the switching mechanism and the tail index of the stable components creates a multi-parameter problem where existing interpolation techniques between Gaussian and stable approximations are insufficient.