I know for sure it doesn't progress while you're logged off, cuz I kill maxed Bee and LightMantis as my Supers, and they kill the bosses in around 2~3 hits when I log on again.
Anyways, hopefully we'll be able to progress while offline later on, cuz as is, I'm barely progressing. -- Reason being webgl doesn't work well w/ my comp, and it crashes if I keep the browser up too long.