Controversial Rectangles

Define an alternating rectangle $R_A$ to be a set consisting of four cells with alternating digits that occupy exactly two rows, two columns and two subgrids. As shown below, there are no rectangles in Figure 1, but some in Figure 2.

Figure 1. There are no $R_A$ 's in this 9-by-9 sudoku. Figure 2. There are 8 $R_A$ 's in all for this 4-by-4 sudoku since the digits alternate respectively within rows and columns. One of the possible rectangles contains A3, A4, D3 and D4.

Is it possible to fill all blanks in 4-by-4 sudoku without creating any $R_A$ ?

Note: For 4-by-4 sudoku, each and every subgrid, column and row consist all distinct digits from 1 to 4.

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Harder version.