CanCPInvariance be Saved?

We examine the experimental basis for the generally accepted statement that lingering $2\pi$ decays from a neutral kaon beam imply that $\mathrm{CP}$ invariance must be abandoned. We find that, apart from formulations which cannot be distinguished physically from theories admitting $\mathrm{CP}$ noninvariance, no $\mathrm{CP}$-invariant description seems possible within the context of quantum theory. If quantum mechanics is disregarded, $K^0$-decay experiments do not necessarily contradict $\mathrm{CP}$ invariance. A crucial test is proposed for that case.