This method will not work. My counterexample consists of 5 points, 4 red, and 1 blue, shown below. I also show that for the lower-left red point, it has two GG neighbors that are also red, so it will be discarded. The blue point is not a GG neighbor.

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Now I show that for the upper-left red point, it has 3 GG neighbors that are red, and the blue point is not a GG neighbor, so the upper-left red point will also be discarded. By symmetry the same applies for the lower-right red point. Trivially, we can see that no other points are discarded.

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So our set X_G consists of the 2 red points and 1 blue point at the top of the following image. Let us query the original lower-left red point, now shown black.

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We see that the blue point is the closest, so the red point will be classified as blue, incorrectly.