It is also logically flawed. If there exists in this world someone who was truly objective. Would they know they're objective?
Knowing if oneself is objective or not is in itself an objective calculation. If that person was truly objective, then the answer is yes, he would know. But I get your point, there are many who think they are "objective" but they are not.
Mind you, one test for objectivity is to see whether your thoughts are the result of parroting cultural norms or if they arise from objective thinking. If your thoughts are objective there is no rule that such thoughts should imitate cultural norms. One will notice that ones opinions are "different" so to say.
My points pass those tests. It doesn't mean I am "objective" but it is evidence for it.