Is it possible to stay signed into sync while signing into different Gmail accounts?

[Jasper]

I run an educational computer lab and it was always a huge benefit of pre-"Lets change everything up" Chrome to allow users to bookmark sites while logged into a central Chrome account but still be able to log into their respective Gmail accounts. I have an auto-log out extension that works like a charm but didn't affect the sync account, every time a user fired up Chrome it remained synced but they had to log in to their Gmail (policy is to never leave personal data on the screens but users forget).

I'm experimenting with Cent on my personal PCs and so far I like what I see but is this possible even with an extension?