Crampons would be better for going up the chute.
Look up videos on how to glissade. You NEED to know 1. how to brake using an ice axe & 2. do not wear crampons while glissading.
By late June, if people are still glissading, the chute will be a dee[, narrow, icy trench. I wouldn't want to try that myself. Walking down would be way safer.
