The question is the very last sentence at the end of this post. In this post, I'll demonstrate how I reach to a contradiction(the conditions mentioned in conjecture 1 should be satisfied by all stationary points in the phase space but I've derived some stationary points that do not satisfy the conditions. So I wonder: How can resolve this contradiction?).
A bead is threaded on a friction-less vertical wire loop of radius $R$. The loop is spinning w.r.t. a fixed axis shown in the figure at a constant angular speed $\omega$. The Lagrangian is given by
$$L=\frac{1}{2}\dot\theta^2R^2+\frac{1}{2}R^2\sin^2\theta\omega^2+gR(\cos\theta-1)$$, where $\theta$ is defined in the figure(the arrow near the character $\theta$ indicates the direction in which it increases).
The equation of motion:
$$\ddot\theta=\sin\theta\cos\theta\omega^2-\frac gR\sin\theta$$
Next let's find out the stationary points in the phase space(A space with two degrees of freedom in the values of $\theta$ and $\dot \theta$. In addition, stationary points are points in the space which if we set as the initial condition would make the bead stay at where it is initially. e.g. ($\theta=0$, $\dot\theta=0$)): $$\ddot\theta=0\implies\sin\theta(\cos\theta\omega^2-\frac gR)=0\implies\sin\theta=0 and/or,\cos\theta=\frac g{R\omega^2}$$
Hence, any stationary point should satisfy at least one of these conditions(conjecture 1). But here comes a problem:
First, approximating the e.o.m. in the vicinity of $\theta=\frac{\pi}2$, we get:
$$\ddot{\bar\theta}+\omega^2\bar\theta=0$$
where $\bar\theta=\theta-(\frac{\pi}2-\frac{g}{R\omega^2})$. Therefore, $\frac{\pi}2-\frac g{\omega R^2}$ is a stationary point between $0$ and $\pi$. Approximating the e.o.m. around $\theta=\frac{3\pi}2$, we get a similar result: $\frac{3\pi}2-\frac g{R\omega^2}$ is a stationary point between $\pi$ and $2\pi$. Therefore, since the integral curves in the phase space are periodic with period $2\pi$, we see that $\frac {\pi}2-\frac g{\omega R^2}+n\pi, \forall n\in \mathbb Z$. However, these points may not satisfy the conditions mentioned in conjecture 1. Maybe they always satisfy the conditions but I don't think it seems to be the case.
There seems to be a contradiction here. Can somebody please help me resolve it?
Approximation around $\frac{\pi}{2}$ is only valid for a small (technically infinitesimally small) deviations, which implies that the result is only valid when $\frac{g}{R\omega^2}$ is very small. This means that $\cos\theta = \cos(\frac{\pi}{2}-\frac{g}{R\omega^2}) = \sin\frac{g}{R\omega^2} \approx \frac{g}{R\omega^2}$, pretty close to what conjecture 1 states.
So it is a good approximation indeed.