Sudoku Conjecture about rows and columns

Question

In the game of Sudoku, if all the columns and rows are properly filled, will that assure all the boxes will be properly filled?

Answer 1

Counter example: $$\left[\begin{array}{ccc|ccc|ccc} \color{blue}{1} & \color{blue}{2} & \color{red}{3} & 4 & 5 & 6 & 7 & 8 & 9 \\ \color{blue}{2} & \color{red}{3} & \color{blue}{4} & 5 & 6 & 7 & 8 & 9 & 1 \\ \color{red}{3} & \color{blue}{4} & \color{blue}{5} & 6 & 7 & 8 & 9 & 1 & 2 \\ \hline 4 & 5 & 6 & 7 & 8 & 9 & 1 & 2 & 3 \\ 5 & 6 & 7 & 8 & 9 & 1 & 2 & 3 & 4 \\ 6 & 7 & 8 & 9 & 1 & 2 & 3 & 4 & 5 \\ \hline 7 & 8 & 9 & 1 & 2 & 3 & 4 & 5 & 6 \\ 8 & 9 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 9 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\end{array}\right]$$

Every box is not properly filled as the anti-diagonal has identical element.