Screw the cap tightly onto the bottle and squeeze the sides. The eyedropper should drop to the bottom, if not adjust the quantity of air in the eyedropper. As you squeeze the bottle the volume inside is decreased, you compress the air in the eyedropper and in the top of the bottle.
Water is forced into the eyedropper.
The eyedropper now has more mass, more density and sinks. Adjust the water levels in eyedropper and in the bottle.