Belt and Pulley Devices, the Simple Answer.
The solution is simple. Consider the vertical sections of belt. Each of these has 5 rods and 5 sliding weights. The weights are equal on the left belt segment and the right belt segment. Therefore the upper pulley supports the same weight on its left and right sides, and there's no net torque to turn the pulley. This is true no matter where each weight is on its rod. End of story.
But you still may be bothered about the apparent torque imbalance that may have initially distracted you from seeing this solution. You need to do a proper force and torque analysis using free-body diagrams.
Consider a belt made up of short rigid sections linked together, with a rod rigidly attached to each, and a sliding weight on the rod. Now, let's break this system down by looking at just two of these, one on the side of the belt which allows the weight to slide to the far end of the rod well away from the belt, and one on the other side, where the weight rests near the belt. [Fig 2-1] The belt links tilt, and the gravitational torque on the weight is balanced by the couple due to the belt tension. [Fig 2-2] The belt sections above and below the link are now a bit off vertical, and this causes horizontal comonents of force at the upper and lower pulleys, where the belt contacts the pulleys. [Fig 2-3] And, in turn, the pulley bearings counter these forces with two horizontal forces, forming a couple. But remember, horizontal forces at the bearings cannot cause the pulleys to rotate. So there is more torque on the right side, but this is balanced by the torque due to horizontal forces at the pulley bearings. The net torque on the system is zero. [Fig 2-4]
This is the process of analyzing a system by considering free-body force diagrams on each of its parts. We have shown only the parts we need to examine, and the figures are schematic only, the forces not being drawn to scale. In particular, we have greatly exagerrated the size of the kink in the belt.
This analysis considered just one sliding weight. In a belt with many of them things get a bit more complicated to draw, but the conclusion is the same. If the belt with one weight has no tendency to turn, the belt with N weights won't turn either.
Once you do this analysis, thinking through the force and torque analysis, you then wonder "How could this have ever looked the least bit plausible as a perpetual motion machine?" Then you realize how seductive these devices can be when only casually considered.
It also becomes clear that this pulley device (and most other belt and pulley perpetual motion machines) are nothing but stretched versions of the early overbalanced wheels, which also didn't work. Adding the belts gains nothing but additional friction.
But, to the average person there's something persuasive about the belt, and the fact that on the vertical segments the system obviously has weights continually extending out farther from the pulley axle. This leads inexperienced people to suppose that that there is a torque imbalance that must cause the system to turn continually.
The experienced engineer or physicist goes right to the simplest and most reliable method of analysis. Look at the work done as the weights move vertically. The weights are equally spaced along the belt, so there are always the same number of them on either side of the system. And in a given time, the number moving up equals the number moving down, and they move the same distance. So the net work in moving the system is zero in any given time. The distance of these weights from the axles is simply irrelevant, since whatever this distance the weights move the same distance vertically. This also allows us to conclude that lengthening the belt won't help at all, for N x 0 = 0 whatever the size of N.
Thought problem. Has this any functional similarity to the Roberval balance?