My intuition on the problem is the following:
Consider a rotating horizontal bar. If there is no friction along the bar, and since the ring has no vertical movement, the only horizontal force applied by the bar to the ring has to be perpendicular to the bar in the direction of rotation. This force has to exist, since the ring is not moving with a constant velocity vector.
Since in this case, there is no inward force along the axis of the bar, the distance to the centre of rotation is increasing.
If the bar is inclined, we have to consider the balance between effect of the force the bar is applying to the ring (that increases the distance to the origin) and the gravity projected along the axis of the bar (that pulls the ring to the origin). If the angular velocity and the distance to the centre of rotation are large enough to overcome the effect of the gravity, then the ring will slide outwards, otherwise, it will slide inwards.
In more detail, lets assume that the bar starts at the origin $(0,0,0)$.
In Cartesian coordinates, the position $\vec{r}$ of the ring at time $t$ when it is at a distance $d(t)$ from the origin, is given by $(x, y, z)$, with:
$$
\begin{align}
x(t) & = d(t) \sin \alpha \ \cos(\omega t + \phi)\\
y(t) & = d(t) \sin \alpha \ \sin(\omega t + \phi)\\
z(t) & = d(t) \cos \alpha
\end{align}
$$
The velocity $\dot{\vec{r}} = \vec{v}$ is:
$$
\begin{align}
\dot{x}(t) & = \dot{d}(t) \sin \alpha \ \cos(\omega t + \phi)
- \omega \ d(t) \sin \alpha \ \sin(\omega t + \phi)\\
\dot{y}(t) & = \dot{d}(t) \sin \alpha \ \sin(\omega t + \phi)
+ \omega \ d(t) \sin \alpha \ \cos(\omega t + \phi)\\
\dot{z}(t) & = \dot{d}(t) \cos \alpha
\end{align}
$$
And the acceleration $\ddot{\vec{r}} = \dot{\vec{v}} = \vec{a}$ is:
$$
\begin{align}
\ddot{x}(t) & = \ddot{d}(t) \sin \alpha \ \cos(\omega t + \phi)
- 2 \ \omega \ \dot{d}(t) \sin \alpha \ \sin(\omega t + \phi)
- \omega^2 \ d(t) \sin \alpha \ \cos(\omega t + \phi)\\
\ddot{y}(t) & = \ddot{d}(t) \sin \alpha \ \sin(\omega t + \phi)
+ 2 \ \omega \ \dot{d}(t) \sin \alpha \ \cos(\omega t + \phi)
- \omega^2 \ d(t) \sin \alpha \ \sin(\omega t + \phi)\\
\ddot{z}(t) & = \ddot{d}(t) \cos \alpha
\end{align}
$$
In order to simplify the analysis, let's assume that $t$ is such that $\cos(\omega t + \phi) = 1$ and, thus, $\sin(\omega t + \phi) = 0$.
Then the acceleration would be:
$$
\begin{align}
\ddot{x}(t) & = \ddot{d}(t) \sin \alpha
- \omega^2 \ d(t) \sin \alpha\\
\ddot{y}(t) & = 2 \ \omega \ \dot{d}(t) \sin \alpha\\
\ddot{z}(t) & = \ddot{d}(t) \cos \alpha
\end{align}
$$
Simplifying once more, let's start with the case where the bar is horizontal. In this case, $\sin \alpha = 1$ and $\cos \alpha = 0$ and the acceleration becomes:
$$
\begin{align}
\ddot{x}(t) & = \ddot{d}(t) - \omega^2 \ d(t) \\
\ddot{y}(t) & = 2 \ \omega \ \dot{d}(t) \\
\ddot{z}(t) & = 0
\end{align}
$$
If we think about the system in this case, the gravity will be ignored, because it will be cancelled out by the vertical reaction of the bar.
Regarding the horizontal force the bar is applying to the ring, we can imagine that is along the $y$ axis and is zero along the $x$ axis. It is zero along the $x$ axis because I'm assuming there is no friction in this direction.
So we can obtain $d(t)$ by solving the acceleration equation along the $x$ axis:
$$
0 = \ddot{d}(t) - \omega^2 \ d(t)
$$
Therefore we get:
$$
d(t) = k_1 e^{\omega t} + k_2 e^{-\omega t}
$$
for some constants $k_1$ and $k_2$.
If we go back to the inclined bar, we may consider an inclined coordinate system that, at time $t$ has the $x$ axis aligned with the bar.
In this case, we have to consider the component of the gravity along this $x$ axis and the acceleration along this axis is:
$$
- g \cos \alpha = \ddot{d}(t) - \omega^2 \ d(t)
$$
Which results in:
$$
d(t) = k_3 e^{\omega t} + k_4 e^{-\omega t} + \frac{g \cos \alpha}{\omega ^ 2}
$$
for some constants $k_3$ and $k_4$.
