= Solution
For $|x|\ll s_0$ the quadratic <Hamiltonian> describes a <simple harmonic oscillator> with
$$
\omega_s^2=\frac{3g\cos\theta}{s_0},\qquad
\omega_\phi^2=\frac{l^2}{m^2s_0^4\sin^4\theta}=\frac{g\cos\theta}{s_0\sin^2\theta},\qquad
\boxed{\frac{\omega_s}{|\omega_\phi|}=\sqrt3\sin\theta.}
$$
An exactly circular <orbit> closes for every nonzero $l$. A nonzero infinitesimal radial <oscillation> closes at leading order precisely when $\sqrt3\sin\theta$ is rational. This condition concerns the cone angle: changing $s_0$ or $l$ cannot change the leading-order ratio. Thus there are no special nonzero $l$ values selecting closure for a fixed cone at this order.
At finite radial amplitude the system is equivalent, after unrolling the cone with planar angle $\chi=\phi\sin\theta$, to a planar <central force> with potential proportional to $s$. Its angle advance per radial period changes continuously from $2\pi/\sqrt3$ near a circle towards $\pi$ at very large radial energy. Hence the physical advance changes from $2\pi/(\sqrt3\sin\theta)$ towards $\pi/\sin\theta$. This nonzero interval contains rational multiples of $2\pi$, so tuning a larger initial $|p_s|$ can produce closed <orbits> even when the small-amplitude ratio is irrational. Generic amplitudes still produce a <precessing orbit>; neither energy conservation nor increasing $|p_s|$ guarantees closure.
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