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If you have to constantly take the social initiative, then you're carrying the relationship.I don't think that's true, and the article does talk about this a bit. Relationships are never actually 50/50. If you're expecting that, you're gonna have a bad time.
But really, think about outcomes more than trivial details: if you are the one who always has to be the one to reach out, but you end up having a good time when you talk or meet up with the person, then isn't that still a good outcome?
But ultimately everyone gets to make their own choice, of course. Would you rather have no friends or family, or would you rather have to put in more effort than the other person at some aspects of maintaining a friendship/relationship? Ideally it shouldn't be this stark dichotomy, but if it is, you gotta make your choice.
(Also, "no unsolicited advice please" is a weird thing to say when you're posting about a personal situation on a public forum. If you don't want to take this as advice, then take this as disagreement with what you're saying, and my expression of frustration that your choice seems to be "be alone and sad" when the alternative is "put in a little more effort than the other person".)