I don't think it's quite about competition per se, so much as it's about expectations. These are obviously not entirely distinct, but I think expectations is a much better way to frame what they imagine in lesbian relationships. Since, the fixation is usually on how straight women have expectations that they weren't 'made for' or some such nonsense. Importantly, this is not just about getting relationships (where competition matters), but also keeping or being happy in them.
The fantasy they construct is that lesbian relationships don't require them to do or be whatever; that somehow in a "lesbian" relationship they would just be more compatible for some mystical reason. I don't really think the thought of other men quite enters into it except to validate the delusion that they're actually different, better even, and thus a "lesbian".
From twitter:
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