Would not bet the farm on this info, but per the DoD instruction number 6130.03 "Medical Standards for Appointment, Enlistment, or Induction in the Military Services" dated 28 April 2010, following is an excerpt pertaining to asthma. My DS followed this after he received a request for more info from the DoDMERB, and ended up being qualified.
Unless otherwise stipulated, the conditions listed in this enclosure are those that do NOT meet the standard by virtue of current diagnosis, or for which the candidate has a verified past medical history.
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11. LUNGS, CHEST WALL, PLEURA, AND MEDIASTINUM
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d. Airway hyper responsiveness including asthma (493.xx), reactive airway disease, exercise-induced bronchospasm (519.11) or asthmatic bronchitis (493.90), reliably diagnosed and symptomatic after the 13th birthday.
(1) Reliable diagnostic criteria may include any of the following elements: substantiated history of cough, wheeze, chest tightness, and/or dyspnea which persists or recurs over a prolonged period of time, generally more than 12 months.
(2) Individuals DO MEET the standard if within the past 3 years they meet ALL of the criteria in subparagraphs 11.d.(2)(a)-(d).
(a) No use of controller or rescue medications (including, but not limited to inhaled corticosteroids, leukotriene receptor antagonists, or short-acting beta agonists).
(b) No exacerbations requiring acute medical treatment.
(c) No use of oral steroids.
(d) A current normal spirometry (within the past 90 days), performed in accordance with American Thoracic Society (ATS) guidelines and as defined by current National Heart, Lung, and Blood Institute (NHLBI) standards.
So we went to a asthma/allergy specialist (I have real good insurance) who checked him out, to include a spirometry (which he passed without issue), and after the Doc reviewed the remed questionair and the DoD standards wrote a note in his file (and addressed to the DoDMerb evaluating him under the above conditions).