Answer
A pilot diameter (either a shaft or a bore) is necessary to prevent the wave spring from shifting off-center, which would cause eccentric loading. For an internal bore pilot, the spring $O.D.$ should be slightly smaller than the bore $I.D.$ at the work height. For an external shaft pilot, the spring $I.D.$ should be larger than the shaft $O.D.$ at the free height. If the pilot is too tight, it will restrict the radial expansion mentioned in Q77, leading to 'binding' and premature fatigue failure due to friction-induced heat.