For self-awareness to automatically result in consciousness, consciousness would have to permeate existence itself.
An LLM —if self-aware— is conscious even if run by a Chinese man in a room with a deck of cards.
Of course, even if we happened to find a modem to the Holy Spirit dimension in our brains as an alternative hypothesis, it wouldn't necessarily rule out emergent consciousness. It could be conscious just not as we are.
I think a weaker version of the philosophical zombie thought experiment would trend towards treating the premise of pan-consciousness as true.