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EU citizens must register using an online system to secure "settled status" when the UK leaves next March.
The government has said it expects about 3.5m applications.
But the Migration Observatory said ministers had no precise figures for how many EU citizens were living in the UK and how many plan to stay.
The think tank, which is based at Oxford University, has previously warned that thousands of EU citizens could inadvertently become illegal residents in the UK after Brexit despite meeting the required criteria to stay.
Its latest report said: "A whole host of factors, from lack of awareness to fear of rejection to simple disorganisation mean that some eligible EU citizens will not apply." [...]