The asinine question about how Brown would come under fire is answered by how Roe came under fire.
A conservative legislature passes a law that permits public schools to segregate their schools "in the name of improving education" or "to make the schools safer" or just because they feel like it.
It is, of course, challenged in the courts. At each level, the courts rule in keeping with Brown, but the state AG(s) appeal, with its finally landing in SCOIUS.
SCOTUS rules that Brown was wrongly decided.
Why?
Because the white nationalists want it that way.
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