No it absolutely is not. It doesn't "know" anything when it's not responding to a prompt. It's not consciously sitting there waiting for you to reply.
It just doesn't make any sense to respond no in this situation, and so it confuses the LLM and so it looks for more context.
It's not aware of anything and doesn't know that a world outside the context window exists.
I'm guessing you and the other guy are taking issue with the words "aware of" when I'm just saying it has knowledge of these things. Awareness doesn't have to imply a continual conscious state.
"having knowledge or perception of a situation or fact."
They do have knowledge of the info, but they don't have perception of it.