In my experience, this does not work.

I gave up on shampoo for about a month and a half, but I still used conditioner. I never used soap on my body when showering to begin with, except for ahem the bottom area.

It actually worked for a little while; the conditioner dissolved the grease out of my hair (like dissolves like). But then for about a week my hair was determined to stay greasy and no amount of conditioner would get it out. So I gave up and started using shampoo again.

There's a reason that in the old days everyone wore hats or bonnets regularly.