I am not sure how to define consciousness, but I can't imagine a definition that doesn't involve state or continuity across time.

It sounds like we're in agreement. Present-day AI agents clearly maintain state over time, but that on its own is insufficient for consciousness.

On the other side of the coin though, I would just add that I believe that long-term persistent state is a soft, rather than hard requirement for consciousness - people with anterograde amnesia are still conscious, right?

Current agents "live" in discretized time. They sporadically get inputs, process it, and update their state. The only thing they don't currently do is learn (update their models). What's your argument?