I sent an email to Alfred Geroldinger with a link to this thread. Here is a summary of his reply (I'm posting with his permission):

 - The structure of minimal zero-sum sequences of maximal length over $C_3^n$ is unknown.
 - Lemma 4.2 in [Geroldinger and Schmid, J. Korean Math. Soc. 56 (2019), No. 4, 869-915. DOI: [10.4134/JKMS.j180467][1]] gives the structure of minimal zero-sum  sequences of maximal length over Abelian groups of rank two.

UPDATE (2024-08-29). As I've just learned from Alfred Geroldinger, the structure of minimal zero-sum sequences of maximal length over $C_3^3$ is the subject of Lemma 5.4 in [Geroldinger, Grynkiewicz, and Schmid, J. Théorie Nombres Bordeaux 23 (2011), Nr. 1, 137-169]. This answers a question by Mikel Martinez Puente in the comments on this post.


  [1]: https://jkms.kms.or.kr/journal/view.html?doi=10.4134/JKMS.j180467