The trail at the 99 switchbacks will likely be inaccessible, so people will be climbing "the chute" (the steep snow slope just west of the switchbacks--not a chute). There will probably be some going without ice axes, but if the snow conditions are such that there is a hard icy crust, an ice axe (and knowledge/experience on how to self-arrest) could be a lifesaver.
If the snow is softer, hiking poles would be ok. Understand that in the later afternoon, soft snow hardens up when the sun leaves that slope and temps drop.
Under NO circumstances should you attempt to glissade (sit on your butt and slide) down that slope without an ice axe as a brake, OR while wearing crampons!!!