Is eosinophilic bronchitis a type of hyper eosinophilic syndrome or are they interrelated?

No. We are still trying to find out what this means although there is suggestion that this may well be a precursor of asthma. It is not related to hypereosinophilic syndrome per se. Once you developed allergic asthma, eosinophils are often elevated in the airway and it is no longer a case of eosinophilic bronchitis but a case of eosinophilic inflammation.