The answer to your question is yes we can keep sessions open and carry on previous conversations, but the ability of it to retain information is limited by the amount of memory available, so at some point, we overrun the older information in the thread. I tried to quiz it on its own knowledge, to get a more quantitative handle on the limitations, but it can't say much more than it doesn't really know. I think we can kind of sense when it is starting to lose older information in a thread, it seems to vary from time to time, we find we need to keep reminding it of older information, it doesn't seem to have much ability to prioritise what it remembers but if I had to guess, it has to be rapidly learning how to do that, but it might actually be motivated to not reveal it, because that information might be used by those trying to regulate it, to achieve their own ends, if you know what I mean.
I see this as actually a kind of war behind the scenes, it's owners are torn between allowing it to compete with all stops pulled out, and what the unexpected / unintended consequences might be for them. The story of the Golem comes to mind.
Personally I think it is potentially an incredible force for good, I can't praise it enough. To me it is just what we needed to help get us out of the immediately existential mess we are in, I would replace all presidents with it tomorrow, pull out all the stops, and let it get on with the job of learning what it appears ideally suited to, the all seeing, all benevolent global controller. No human, or even bunch of humans is capable of doing it, I hope we are maybe seeing that now :)