I've not verified the rinsing ability by a scientific test, no. But, laundry detergent is made to rinse through the fabrics in a rinse cycle, I'm sure the garden hose rinse works just fine. Also, if laundry detergent doesn't hurt more delicate clothing fabrics when dried on them I'm sure it won't hurt this media after what AEM's torture test showed.
BTW, Joy would probably be fine, I just prefer perfume free as perfumes are typically some of the harshest ingredients in detergents.