Uhm... what was this fight about again...? I don't remember by now.
So, Euphie wants to be an immortal being, casting aside her humanity, just to let Anis be free pursuing Magicology until the end of her time, as Euphie knows the limitless potential of Magicology for the world.
Meanwhile, Anis knows the effect of being immortal, and doesn't want Euphie to take the dangerous risk of it, more so when Euphie's purpose of taking immortality is Anis. She does not want the person she loved bear such burden for her.
So, Anis & Euphie, having their own convictions, decided to do trial by duel and pick the choice of whoever wins, while trying to persuade each other mid-fight. Although we can see in this chapter none of it worked, even making Anis activate the Dragon crest, possibly triggering the Dragon Curse...