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Can the CP asymmetries in $B\to ψK_S$ and $B\to ψK_L$ differ?

In the standard model the CP asymmetries in $B\to ψK_S$ and $B\to ψK_L$ are equal in magnitude and opposite in sign to very good approximation. We compute the order $ε_K$ corrections to each of these CP asymmetries and find that they give a deviation from $\sin 2β$ at the half percent level, which may eventually be measurable. However, the correction to $a_{CP}(B\to ψK_S)+a_{CP}(B\to ψK_L)$ due to $ε_K$ is further suppressed. The dominant corrections to this sum, at the few times $10^{-3}$ level, come from the $B$ lifetime difference, and CP violation in $B- \bar B$ mixing and $B \to ψK$ decay. New physics could induce a significant difference in the $\sin(Δm_B t)$ time dependence in the asymmetries if and only if the "wrong-flavor" amplitudes $B \to ψ\bar K$ or $\bar B \to ψK$ are generated. A scale of new physics that lies well below the weak scale would be required. Potential scenarios are therefore highly constrained, and do not appear feasible. A direct test is proposed to set bounds on such effects.