Of course, it is possible that you could have some other type of feedback loop that produces self awareness and consciousness, but from what we understand of how LLMs work, it seems highly unlikely that statistical token generation is sufficient for that.
This is where "agentic" models enter the chat. A coherent identity and goal, provided by the user adds an additional layer above the statistical-generation of simple LLMs. This layer can be engineered for persistence, even through malicious means like prompt injection.
As an academic exercise, I propose that some persistent-subset of these AI agents possess the "embodiment" we believe is missing. If all this amounts to is a Proof By Negation? Good, we get somewhere that proves our current AI is not conscious.