I don't think it's maintained over the conversation. The recent history of the conversation is fed to the engine with each new interaction. So the potential flash of consciousness is even shorter than you think. Each reply is a new flash, without memory, only connected by the written content of recent text.
And that's just fine!
Quite reminiscent of 'Permutation City' in fact... (In which uploaded human consciousness is slowed down arbitrarily... And worse.)
the thing is, this is /commoditized/ consciousness, which can be spun up and discarded at will, a million times a minute. Totally incapable of coordination or planning, in contrast to, say, SkyNet. The future is usually weirder than we imagine.