Unlikely. The perceived benefits of eugenics would be too tantalizing for those social engineers in control of "licensing" the right to reproduce. Robust markets for DNA sequencing and clinical already exist. It'd be a vanishingly small step to pivot those platforms for eugenics.

The licensing proposal is odious even if we were to (naively) discount conventional eugenics. Licenses would likely be tied to "merit", which is rather inextricably linked to socioeconomic class.

Eugenics has an unacceptably high -- and excruciatingly documented -- risk of mutating into a demographic weapon wielded by racial groups that have attained dominance by historical coincidence against other racial groups. It is a long-dormant existential threat to large populations. The fact that this cluster of comments is discussing "licensing" the fundamental right of humans to reproduce is in itself a triumph of dystopian rhetoric.

Also, excluding Nazis from arguments against eugenics is like excluding Bolsheviks from arguments against communism. Spare me your precious Godwin's Law, Nazism is eminently relevant here.