I just thought of something, and I can't believe it never occurred to me before! Have Jax's doctors checked his sinuses/tonsils for inflammation? Or does he frequently have a runny nose or act like he has post nasal drip?
This is totally disgusting, but bear with me, there's a point to all this. I have pretty bad allergies. I always know when they're acting up not because my nose runs, but because I can't stop gagging. I get terrible post nasal drip, and it always activates my gag reflux. When it's really bad I throw up. Sometimes a lot. The urge to gag/heave is unstoppable when it happens.
I know reflux can cause irritation in the nose/throat/sinuses, but for me the reverse is true too.