Is there really always at least one prime number between n squared and (n plus one) squared, for every positive integer n?

No proof or counterexample has ever been found, despite the claim being checked by computer for enormous ranges of n. Results exist for closely related but weaker statements, such as a prime between consecutive cubes for large enough n, but nobody has found a way to close the gap for consecutive squares, which is exactly the gap that makes it one of Landau's four unapproachable problems named in 1912.