Letīs consider it this way:
So long as the engine thrust is flat rated at a fixed force
the rudder moment is independent of density altitude at any fixed IAS
thus the control speeds in terms of IAS are independent of density altitude, while control speeds in terms of TAS increase with density altitude.
When the engine thrust is not flat rated and the force derates with density altitude
the control speeds in terms of IAS decrease with increase of density altitude
while the behaviour of control speeds in terms of TAS depends on how fast thrust derates.
Does anyone know how the TAS control speeds depend on density altitude for non-flat-rated engines?