I have had an inspiration of how to solve this question. The conclusion might seem shocking and counter intuitive to some, but I only ask you carefully read the arguments presented below, before judging.
Argument 1) First lets us consider a lab in an elevator that is initially at 'infinity' from a gravitational body. There is a thin horizontal rod that is spinning around its central vertical axis and an observer with a clock in the elevator. For the sake of argument, let's say the tips of the rod are measured to have a tangential velocity of 0.6c and the observer times the rod to complete a revolution once per second. The elevator is lowered close to the gravitational source, to a height where the gravitational gamma factor is 2 according to the observer at 'infinity'. If the spin rate of the rod does not slow down according to this observer, the observer in the lab will measure the tangential velocity of the spinning rod to be 1.6c, due to gravitational time dilation of her clock. Obviously this result cannot happen in the context of relativity, so we can only conclude that the spin rate of the rod must slow down in the same way as the lowered clock. The lab observer will measure the spin rate to be constant all the way down and angular momentum is conserved from their point of view. The spinning rod (or any object with angular momentum with no external forces acting on it) is in fact a primitive clock in its right.
Argument 2) Now we extend the height of the lab to several tall stories. There is a thin flywheel rotating about its vertical axis at the top room and a similar flywheel at the bottom room. The two flywheels are connected by a transmission rod. In the middle room a clutch assembly is inserted in the middle of the transmission rod and is initially engaged. The mass and moment of inertia of the connecting rod and clutch is intended to be negligible compared to those of the flywheels. Initially at 'infinity' the flywheels are both spun up to 4 rpm. Observers with their own clocks are placed in each room and confirm the flywheels and the two clutch plates are all rotating at the same rate by their own measurements. The clutch plates are disconnected and are observed to continue to rotate at the same rate initially.
The lab is now lowered into the gravitational field. As it descends the top and bottom observers note that their local flywheel maintains a more or less constant angular velocity as they descend. Observer B in the middle notes that the two clutch plates increasingly rotate relative to each other as they descend. The clutch plates have negligible angular momentum compared to the flywheels and effectively just follow the rotation rate of the flywheel they are connected to.
When we reach a point in the gravitational field where the gamma factor for the top observer is 2 and the gamma factor for the bottom observer is 4 according to the observer at 'infinity' we stop. The top and bottom observers measure the rotation rates of their flywheels to be more or less the same as when they started, but the observer at infinity now measures the top flywheel to be rotating at 2 rpm and the lower flywheel to be rotating at 1 rpm. Now we engage the clutch in the middle. The result is that the top flywheel slows down according to the observer at the top and the lower flywheel speeds up according to the observer at the bottom. There will be shear stress in the connecting rod, but as long as it does not break, the two flywheels will reach an equilibrium with an angular velocity $1 < \omega <2$ rpm according to the observer at infinity. The observer at the top will see their flywheel rotating at significantly lower than 4 rpm and the lower observer will see their local flywheel rotating at significantly greater that 4 rpm.
Now we conceptually simulate the cylinder by replacing it with a stack of thin flywheels all rigidly connecting to each other. Following the arguments above, the local observers that remain near the top flywheels, see them slow down as the stack is lowered and the corresponding observers near the lower flywheels, see them speed up continuously as the stack is lowered. This is the answer to the original question if we replace the stack with one solid cylinder.
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The remaining part of this answer just fleshes out the arguments presented above:
Appendix A)
In argument 2 we can make an analogue of this effect by having two electric motors with flywheels that are initially separate with one rotating at 2 rpm and the other rotating at 1 rpm. If we have sandpaper on the faces of the flywheels to make a primitive clutch and push them together, both flywheels will end up rotating at the same 'average' rate, even though one motor continually tries to accelerate the assembly and the other motor continually tries to decelerate the assembly.
Appendix B)
We can alternatively analyse the situation by considering the equivalence principle. Let's consider a observer accelerating with a constant proper acceleration of 1 g in flat spacetime. This observer is effectively a Rindler observer. He has a clock and an ideal spinning flywheel. As he accelerates, he measures no change in the spin rate of the his flywheel. We know this, because by the equivalence principle, the situation is equivalent to a flywheel on the surface of the Earth experiencing a constant acceleration of 1g and we know from experience that the flywheel rotation rate remains constant to a local observer. However, the inertial observer watching the accelerating observer sees the tick rate of the observer's clock and the rotation rate of the accelerating flywheel continuously slowing down at the same rate, confirming the idea that an ideal flywheel is in effect a clock. However, the slowing down of the flywheel is not a violation of conservation of angular momentum. As the angular velocity slows down in accord with the instantaneous gamma factor of the accelerating flywheel, the moment of inertia of the flywheel increases by the same factor, so that the angular momentum remains conserved from every observer's point of view.
Appendix C) In the above arguments it is considered that the total angular momentum of the assemblies always remains constant from the point of view of the Schwarzschild observer at 'infinity'.
Appendix D) When lowering the solid cylinder or connected flywheels, the angular velocities at the top and bottom of the assembly will always remain equal from the point of view of the observer at infinity. However the shear stress in the vertical cylinder or connecting rod will continually increase due to the difference in local angular velocities and eventually the cylinder or connecting rod will break. This is the analogue of the string in the Bell's rocket paradox breaking, despite its length appearing to remain constant to the inertial observer.
