Do you have carpet at home? Would it be the pollens that cause Sean's eczema? We have carpet in the bedroom. My son, daughter and my hubby had very bad allergy in 1 or 2 month ago. They coughed non-stop. One day, the daycare owner called me saying that my son coughed non-stop and he refused to play, just sitting and leaning his head on the kids' sofa then coughed. I took him to the doctor's clinic and the doctor checked him and said that it's allergy. He even needed to give us a respiratory machine to let Spencer inhalt medicine to help him stop coughing. It turned out work for all kids and my hubby.
Then, I bought a air-purifier and a vaccum cleaner that would pick up dust well (normal standard grade of vaccum cleaner just pull the dust and dirt back to the carpet). I found that work, at least, I see more dust and dirt being picked up from the new cleaner and their cough improved after using the air-purifier.
My hubby and my son's doctor both said that the allergy in this year is bad. Since eczema relates to allergy in some sense, control to avoid source of allergy helps to stop eczema. My son still has bad eczema sometimes, especially when it's hot or the air condition is bad. But, we don't need to use medication all the time now.
Maybe you can see if having a different air-purifier or/and vaccum cleaner can help.