So I have looked at this more closely. I didn’t find any option to pay OASI (de: AHV) anything else than what you owe them. There only is voluntary OASI for people leaving Switzerland. As a student you paid your contribution, so those count as contribution years. The 5 years back thing is only for paying missed contributions. Did you find different information?
I also wanted to know if putting money into OASI was profitable in any way. It has become a wiki post.