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The graphics card always handles color correction, but using a proper ICC profile ensures that all the colors stay within the monitor's color range. This can be important if your monitor is 8-bit+FRC and you set 8-bit in your graphics card settings (and your graphics card does not perform dithering). Without a proper ICC profile, certain shades may fall into the range where the monitor activates the FRC module, which can cause increased eye strain. That said, this might not work the same way for all monitors, so it’s a good idea to test and see whether using an ICC profile feels more comfortable for you.
But the biggest problem is getting a correct ICC profile for the monitor, since not everyone has monitor calibration devices and not all monitor manufacturers provide ICC profiles. However, if you have one, it's worth trying both options.