Likewise, the states were free to make laws prohibiting otherwise qualified voters from voting on the basis of sex until the 19th Amendment to the Constitution was ratified.
Likewise, the states were free to make laws prohibiting otherwise qualified voters from voting on the basis of taxes owed until the 24th Amendment to the Constitution was ratified.
Finally, the states were free to make laws prohibiting otherwise qualified voters from voting on the basis of age until the 26th Amendment to the Constitution was ratified. Note that 26A did not eliminate discrimination of voting on the basis of age, but made prohibiting people from voting based on age discrimination uniform thoughout the states.
So it is reasonable to conclude, imo, that the states can prohibit people from voting on the basis of unverified citizenship status. This is because the states have never ratified an amendment to the Constitution which prohibits the states from prohibiting otherwise qualified people from voting based on failure to prove citizenship.
And even though it took me a short time to write this post, let's see how long it takes the Supreme Court to make a decision in this case, not that outcome-driven, activist justices will agree with what amounts to a common sense (please forgive me) interpretation of the Constitution.
I would only add that Article I Section 2 and the 17th Amendment specify that the qualifications of electors to the House and Senate have the same qualifications as electors to the most numerous branch of the State legislatures. If a State requires proof of birth or citizenship to elect state assemblymen, I do not understand how that can be a contested standard for Congressional elections.