Brzozowski's conjecture: Start from any word w, and replace any occurrence uwⁿ x by uwⁿ⁺¹ x or vice versa. Is the resulting language regular? This is easily seen for n = 1, and it has been proven for n ≥ 3. The case n = 2 is still open.

-- JeffreyShallit - 13 Jul 2011