There were some findings in the 1960s suggesting that “Y-bearing” and “X-bearing” sperm cells had different sizes and swimming movements that might affect their fertilization chances, but recent work with better microscopes says the differences are subtle to the point of insignificance.
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1440662/
https://www.frontiersin.org/articles/10.3389/fcell.2019.00388/full
Genetically speaking a typical XY dad should produce 50% X sperm and 50% Y sperm (give or take a few with nondisjunction abnormalities) and if it’s true that they swim equally well, then other differences could be
* sperm survival rates in the vas deferens
* sperm survival rates in the vagina/uterus/fallopian tubes
* sperm getting-through-the-egg-membrane capability
* fetal survival rates
* fetal health
* sex selection by parents
…and that’s all the other hurdles I can think of, on the way to obtaining a birth certificate.
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