You dumped out the water ten minutes ago. The pot is cooled off, and empty. It only had whole eggs in it, still in the shells. Now tell me, do you wash that pot? Get out the sponge and the soap and scrub it like usual? Or do you give it a quick rinse and slam it in the drying rack? No judgement here, only curiosity.
I do because I live in an area where the water is hard as fuck, and will leave a ring of minerals in my pot if I do not wash it. So regardless of any other factors, I do.