part of today's ruling said that Arizona (or any other state) can check the federal registration form against other records (SSN, state tax records, etc) to verify citizenship if the registrant does not provide it.
"...while the NVRA forbids States to demand that an applicant submit additional information beyond that required by the Federal Form, it does not preclude States from 'deny[ing] registration based on information in their possession establishing the applicants ineligibility.'" Opinion page 12
Which is exactly the problem.
The onus to prove/disprove eligibility is on the state rather than the applicant. The State is barred from obtaining information from the applicant and instead must rely on information in their possession.
Meanwhile the foreigner is registered, and will vote.
A vote can not be undone.