A stiff slinky would
Actually, it's not the stiffness so much as the rest-state helix dimensions that matter. The rest-state of a standard new slinky has the loops in contact with each other which means the slinky loops hit each other during the top-drop dynamics. Such a slinky instantly transitions from elastic to inelastic dynamics to form a shockwave.
To have pure wave equation motion, the rest-state of the slinky must be expanded enough so that when the slinky is further stretched by gravity and then released, the internal compression wave can oscillate to either side of the rest-state condition without the loops hitting each other to create a non-linear incompressible shockwave. That is, the largest loop-to-loop stretched state deflection must be less than twice the rest-state loop-to-loop distance.
If you were to take a new slinky and over-stretch it so that the loops permanently deform in the stretched state, then the dynamics of the top-dropped slinky would be very different.