Uniqueness of the maximal function field with the third largest genus in characteristic $3$

Let n be a positive integer and define q=3ⁿ. It is well known that there exists a maximal function field Z₃ over Fq² of genus g₃=q(q-1)/6 and that g₃ is the third largest genus a maximal function field over Fq² can have. In this article we show that any maximal function field over Fq² of genus g₃ is isomorphic to Z₃. This extends previously known uniqueness result for the largest and second largest genera of maximal curves.