An argument my husband decided we needed to fight about
No. See archimedes' principle - an object immersed in fluid displaces it proportionally to its weight.
More importantly, I don't see why you fill the sink at all. If you have a dishwasher, it's pointless, if you wash manually it's inefficient. Dishes should be left with a bit of water until washed to ease the cleaning. Washing by hand, most of the work is scrubbing with a sponge or brush after some moistening and water is only used to clear the soap at the end.
There will be less water if you fill it when dishes are in it. You can test the same principle when you fill two glasses with water- one with ice cubes in it and one without. After filling take the ice cubes out and see how much fluid you have compared to the other glass.
That's why I take soft drink without ice in a restaurant ;-)