I'm not sure how a PPI would help for CHs. In fact, one of the common side effects is severe headaches.
https://www.nyheadache.com/blog-posts/ppis-are-the-worst-but-all-heartburn-drugs-increase-the-risk-of-severe-headaches#:~:text=A new study just published,having migraines or severe headaches.
"A new study just published in Neurology showed that people taking proton pump inhibitors (PPIs) such as omeprazole (Prilosec) and esomeprazole (Nexium) have a 70% higher risk of having migraines or severe headaches."
I take rabeprazole, for GERD; but, started taking it after successfully busting. So, I can't say whether it has had any effect on my CHs. But, I just don't see how it would help.