In all ICC conversions, there has to be two profiles (color spaces) defined. A source and a destination. An untagged document is always assumed to be in some color space even if that assumption is dead wrong. In Photoshop, whatever color space you have selected in the working space popup menus (Color Settings) is the assumption. So if you print an untagged document, Photoshop assumes this is the color space (that’s what the preview is based upon too). So yes, you must either assign the correct profile for the conversion to the print space OR a color space assumption will be made for you. IOW, it does make a difference when you print because two profiles are always necessary.