If you regularly use fast washes that don’t rinse well then yes, it sounds like common sense. It’s only one of many potential triggers though. My dd hasn’t had much eczema at all since we quit dairy and egg, and our washing machine usage hasn’t changed, so wasn’t a factor for us. Perhaps the child in the article is particularly allergic to washing liquid or fabric softener (which as someone with eczema and allergies myself we never use).