I still don't like that this is here.
The intention of this class is to update the list of projects on the Gerrit server once a minute, which very much different than what runUnreviewedPatchSets does, and will be very confusing for future maintainers that this method is here.

And also that it is called once every minute is a waste of the CPU.

Instead this method should be put in a ConnectionListener and registered with the handler as the other(s) listeners.
I understand that you'd like to reuse the cached list of projects and with the current implementation that would be hard to get to in a connection listener since the list probably hasn't been fetched yet when it gets called. But that would, I think, be better fixed by forcing an update just after the connection has been established and before notifying the listeners.