The article specifically mentions that the weight loss was not enough to explain the effect on the migraine: "Importantly, while participants’ body mass index declined slightly (from 34.01 to 33.65), this change was not statistically significant. An analysis of covariance confirmed that BMI reduction had no effect on headache frequency, strengthening the hypothesis that pressure modulation, not weight loss, drives the benefit."

When I developed pre-diabetes symptoms (having to drink a lot - like 5-6l per day, consequently having to pee a lot, when not doing the first: getting headaches), I noticed that cutting on sugar meant (next to other benefits) that the thirst-induced headaches were the first thing that went away. Not the thirst itself (still have it).

However, not an expert on headaches and can't say if it has anything to do with the migraines in the article. But the point that reducing sugar helps in a myriad of ways, stands and is worth repeating.