Desdemona, who is she, and why do we have to "win her love"? Let's leave aside the speculation that she might be the Lady Elite.
This name is typically Italian/Valencian, and famous through W. Shakespeare's play Othello. ( In fact, the name Iago also occurs in the play as the name of the villain. )
I think we should ignore the Shakespearean allusions, and concentrate on the story. True, we know nothing about the lady, but the phrase "win her love" seems to indicate that she may be hostile to us at first. We may have to do one ( or more quests ) for her to prove our worthiness. ( Remember Hooktail? )
And why should she be hostile to us? Here I'm going to present a crazy speculation - Desdemona is human and her love was killed during the Polarian War - BY PIRATES! Maybe by AVERY, HIMSELF! After all we don't know for sure what side he fought on.
So there it is. Have fun debating!

Hmmm that be quite interesting research their lass, quite nice.

- Deadeye Jack Morgan