= Solution
\b[False.] <Adiabatic invariance of an orbital action> preserves actions under sufficiently slow regular evolution, not the ratios of their frequencies. With <action-angle variables>, $\Omega_i=\partial H/\partial J_i$ can change even when $J_i$ does not. A closed bound orbit requires commensurate radial and angular frequencies; rational commensurability is not an adiabatic invariant.
For an explicit counterexample, slowly change the physical potential from Kepler form to
$$
\Phi(r)=-\frac{GM}{r}+\frac12\alpha(t)\omega^2r^2,
$$
with a small, slowly varying positive $\alpha$. The <radial epicyclic frequency> and circular <angular frequency> at a nearly <circular orbit> are
$$
\Omega^2=\frac{GM}{r^3}+\alpha\omega^2,\qquad\kappa^2=\Phi''+\frac{3\Phi'}r=\frac{GM}{r^3}+4\alpha\omega^2.
$$
Initially their ratio is one. For generic nonzero $\alpha$, $\Omega/\kappa$ is irrational, producing the <Kepler-harmonic epicyclic precession>. An initially slightly eccentric ellipse with nonzero radial action therefore becomes a precessing nonclosed orbit, while its radial action remains approximately fixed and its <angular momentum> remains exactly fixed by <spherical symmetry>.
\b[Adiabatic actions do not preserve orbital closure.] Exceptional circular or resonantly commensurate final orbits do not rescue a claim about every initially closed orbit.
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