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Yes, the material stiffness is unchanged but, the resultant stiffness of the bushing is changed by the force applied to the bushing. For example, if you take a cable and stretch it with very little force it will sag much. As you apply more force or tension, the cable will require more force to deflect due to the increased stiffness.
The statement that as the load increases the natural frequency will change is true. This is an inverse perspective.