Mary has a box with special $2\times1$ dominoes. Each dominoe has two red corners and two blue corners, and these dominoes come in two different types: The first type has the lower left and the upper right corner in red, while the second type has the upper left and the lower right corner in red. In other words, the second type is the mirror image of the first type. The box contains $20$ dominoes of the first type and $12$ dominoes of the second type.
Question: Is it possible to tile a standard $8\times8$ chessboard with Mary's dominoes, so that no red dominoe corner touches any red corner of another dominoe?
(The dominoes may be rotated in the tiling, but they must not be flipped over. Two dominoe corners may touch horizontally, vertically, or diagonally.)