It is known that:

(1) Palindrome can be recognized by two-tape TM in O(N) (2) Palindrome can be recognized by one-tape TM in O(N^2)

Question: do we actually have proof that a one-tape TM can't recognize Palindrome faster than O(N^2)?

[I have some handwavy intuition on why it "must" run back & forth over the tape; but nothing that I can formalize into a proof.]

Thanks!