That direction would likely contain weird outcomes if the science progressed, something like "Dogs are barely-conscious due to their pack structure, they have a couple levels of recursive theory of mind but they can't sustain it as deep as we can. But cats didn't have that pack structure, they're not conscious at all." Or, "this person has such severe autism that he cannot fundamentally understand others' minds or what others interpret his mind to be, so we've downgraded his classification to unconscious. He'll talk your ear off about the various cars produced in a golden age between 1972 and 1984, but because he doesn't really know what it means for you to be listening we regard it as sleep-talking."

It also just kind of doesn't sound right. "What happens when we go to sleep? Well, we stop thinking about what others think we think, and we simply accept what they think about us." That doesn't sound like any sleep I experience -- it might describe some of my dreams, but of course dreams are anomalous conscious experiences that happen during sleep so that also misses the mark.

Thesis : The very thing (a brain module) which allows for outside object continuity, that same brain module maintains inside self / personality / identity continuity.

Reasoning :

Evolution found out modelling the outside world is helpful for survival. Some eons later, it figured out modelling yourself (self / agent) modelling the outside world is also helpful. In the outside world, we keep track of continuity of objects through a brain module which hones in on the essence of objects (e.g. tracking a prey or predator.) so that EXACT matching algorithms aren't applied but ONLY approximate ones. As soon as a high enough approximate match (>95%) is found, we "register" it to be an exact match. i.e. the Brain bumps up the confidence level to 100%. This is also the reason why we consider our friend Bob to be the same childhood Bob even though he has different hairstyle, clothes, and other such properties. We don't call Bob who looks different than yesterday as an Imposter. The damage to this brain module could lead us to call Bob today an imposter. This same module also tracks continuity of self in a similar manner. Even though our "self" changes from childhood to adult we "register" "changing self" to be the same thing. i.e. internally we bump up the confidence to 100% when memories, etc. match and provide a coherent picture of the self. A multiple personality disorder is just different stable states of neural attractor states. Continuity is local to a personality but not global and hence transient. This local continuity of information could link up globally, giving rise to coherent single personality.

Capgras Syndrome :

Ramachandran Capgras Delusion Case : https://www.youtube.com/watch?v=3xczrDAGfT4