= Solution
For fixed nonzero $l=p_\phi$, the radial <effective potential> is $V_{\rm eff}(s)=l^2/(2ms^2\sin^2\theta)+mgs\cos\theta$. Its stationary point is
$$
\boxed{s_0^3=\frac{l^2}{m^2g\cos\theta\sin^2\theta},\qquad V_{\rm eff}''(s_0)=\frac{3mg\cos\theta}{s_0}>0.}
$$
The <equilibrium> is therefore stable. A <Taylor expansion> with $x=s-s_0$ and $p_x=p_s$ gives
$$
H_{\rm eff}=\frac32mgs_0\cos\theta+\frac{p_x^2}{2m}+\frac{3mg\cos\theta}{2s_0}x^2+O(x^3).
$$
\b[Qualification:] the printed “any” $p_\phi$ excludes a regular interior <equilibrium> when $p_\phi=0$. In that case $V_{\rm eff}=mgs\cos\theta$ has no stationary point for $s>0$. Its minimum is the nonsmooth apex, where these coordinates and the smooth-cone dynamics fail.
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