We really need to figure out what sleep is for. It seems important.
If I go without enough sleep, my hair and skin start getting greasy, my immune system is weakened, my muscles start feeling tired, etc.
Even if you don’t bathe at all for a few days, or just rinse yourself with water (skip the oil-stripping soap for this experiment), the greasiness of hair/skin fluctuates noticeably throughout the day, and increases substantially after a day or two of insufficient sleep. (That’s right, the greasiness will go up and then subsequently go down, without your doing anything special in between.)
[Side note: it’s pretty disappointing when people make confident sounding assertions about things they clearly don’t know anything about and didn’t bother to do the tiniest trivial bit of study. I find this happens quite commonly even in discussion venues where there’s ostensibly a cultural value placed on reasoning and evidence.]
Of course I'm going to get more greasy after two days of not using soap. I don't see how that would verify anything that you are claiming.