Univalent means not invective. My only thought on this is maybe to use reflection principle? Can someone drop me some hints?
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Your question should be clear without the title. After the title has drawn someone's attention to the question by giving a good description, its purpose is done. The title is not the first sentence of your question, so make sure that the question body does not rely on specific information in the title. – Martin R Oct 01 '19 at 17:01
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The referenced Q&A shows the contrapositive: If $f$ is injective on the boundary of a disk, then it is injective in the disk as well. – Martin R Oct 01 '19 at 17:05