The person who asks for fair treatment is not responsible for making the request entirely comfortable.

Asking to be treated fairly usually creates some friction, and that friction gets quietly charged to the person who asked. They are told they raised it badly, at a bad moment, in the wrong tone, and the conversation slides from the unfairness to their delivery. Some discomfort is simply what happens when something unequal is said out loud. Being reasonable is worth aiming at, but comfort all round is not the standard.

If you have something to raise, be clear and be civil, and then let the awkwardness exist. When somebody raises something with you, resist the pull toward discussing how they said it.