I think this is not his concern.

Afaik the ESTA rules do no make a hard requirement for how long the passport has to be valid, unlike for a visa.

I would assume: Date of entry + 90 days (which is the legal limit you can stay withe the visa waiver agreement).

So if you go in June you should be safe. But I would ask your carrier (BA?), they should know for sure as they need to make sure you are admitted entry in the country you are trying to travel to.