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Twins and DNA tests

As an identical twin and as someone that shamelessly watches the Maury Show this question always lingered in the back of my mind: If twin A fathers a child, and both twins were given a paternity test, would it say that both twins are the father? I know that twins have the same DNA but is expressed differently through the phenotype. But could this test say that both are the father?
gpt
There are small DNA level differences between monozygotic twins that can be tested for - epigenomic changes (environment changing DNA expression) or differences in copy-number variations.

However, most paternity tests look at only a few parts of an individual's genome, and wouldn't be sensitive enough to differentiate the twins. They look at your "DNA fingerprint," choosing genes that are highly likely to vary between individuals. Monozygotic twins would have the same DNA fingerprint.