Can a solved Sudoku game have an invalid region if all rows and columns are valid?

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Given a $9 \times 9$ solved Sudoku game with $3 \times 3$ regions, is it possible that one (or more) of the regions are invalid if all rows and columns are valid (i.e. have a unique sequence of $1-9$)?

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Yes, it can happen that all $3 \times 3$ regions are invalid:

\begin{array}{|ccc|ccc|ccc|} \hline 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 1 \\ 3 & 4 & 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 \\ \hline \end{array}