Chronic Obstructive Pulmonary Disease (COPD): Given the patient's symptoms of cough and shortness of breath (SOB) that have not improved with multiple antibiotics and steroids, and the presence of minor atelectatic changes on the chest X-ray (CXR), COPD is a strong consideration. The lack of significant response to antibiotics suggests a non-infectious cause, and the absence of lobar consolidation or pleural effusion further supports this diagnosis.

Asthma: Similar to COPD, asthma could explain the persistent cough and SOB, especially if the patient has a history of atopy or allergic reactions. The minimal prominence of pulmonary vessels centrally could be indicative of airway disease.
Interstitial Lung Disease (ILD): The linear density and atelectatic changes seen on the CXR could be early signs of an ILD, which might not have been fully apparent on the imaging but could explain the patient's symptoms and lack of response to typical treatments for infections.
Bronchiectasis: This condition involves permanent dilation of parts of the airways and can cause chronic cough and SOB. The presence of atelectatic changes could be secondary to bronchiectasis, especially if there's an element of airway obstruction.

Lung Cancer: Although less likely given the absence of a distinct mass on the CXR, lung cancer can present with non-specific symptoms like cough and SOB. It's crucial to consider this diagnosis, especially in patients with risk factors such as smoking.
Pulmonary Embolism (PE): While the CXR does not show significant findings suggestive of PE, such as the Westermark sign or Hampton's hump, this diagnosis can be life-threatening and requires consideration, especially if there are risk factors for thromboembolism.
Tuberculosis (TB): TB can present with chronic cough and SOB, and the atelectatic changes could be secondary to TB. The lack of response to standard antibiotics increases the suspicion for TB, especially in endemic areas or in patients with risk factors.

Sarcoidosis: This autoimmune disease can affect the lungs and cause symptoms similar to those described. While less common, it should be considered in the differential diagnosis, especially if other systemic symptoms are present.
Eosinophilic Pneumonia: This rare condition can cause cough and SOB and might not respond to conventional antibiotic or steroid treatment. It's characterized by eosinophilia in the lung tissue and could be a consideration in this case.
Lymphangitic Carcinomatosis: This condition involves the spread of cancer to the lymphatic vessels of the lungs and can cause symptoms similar to those described. It's a rare diagnosis but one that should be considered, especially if there's a known history of cancer.