Or is egalitarianism necessarily narcissist?
The narcissist doesn't see others. They see copies or reflections of themselves. Technically speaking they aren't projecting. Rather, they think you are identical to them, so anything which is true of them must also be true of you.
This, likewise, is the egalitarian logic. Everyone is perfectly identical, therefore...
Narcissism is a big problem with humans because it's so close to babby's first empathy. Mirror neurons don't in fact mirror anything; that's narcissist cope. Mirror neurons make you imagine how you would feel and project it onto the other person, as a crude first guess at empathy.
Mirror neurons are closer to projector neurons, and even closer to conformism neurons. If you reject babby's first empathy, "Oh, no, it isn't bothering me," 99% of the time they'll get really angry with you. The idea is socialization. They see you in a situation, then they model how you should react to the situation, then you copy them and are socialized. Your real feelings are irrelevant, as are theirs.
Likewise if there's more than two people it ensures they're all feeling the same thing. Maybe it was for empathy at one point but now it's entirely about enforcement. Unlike cognitive empathy, affective empathy is a form of assault.
Cosmopolitanism encourages narcissism by overloading the social registers. A kid leaves his mother's skirts when he's underloaded, and retreats to them when he's overloaded. The 100 IQ can't even master his local norms, so handling foreign norms or translation functions is right out. The more differences encroach upon him, the less he can avoid them, the further he retreats into himself until he goes full narcissist. The stress becomes trauma. In the end they get triggered by encountering anyone even slightly different. See red hat, go snowflake.
It's clear narcissism, cosmopolitanism, and egalitarianism all reinforce each other. The egalitarian states will model narcissism and thus make its population very ill, for example.