"Most organic compounds in soaps and detergents dissolve in water and end up in sewer systems (20), with negligible amounts emitted from wastewater treatment plants (21)"

I didn't read the full paper, but this is the only mention of soap in there. Soap seems to be okay with respect to air pollution. Mind you, a lot of "bath bars" on the market these days are not soap, but that's a completely different topic.

Perhaps the article about this paper (and unwashed hippies in general) are wrong about soap?

From the abstract:

"A detailed mass balance demonstrates that the use of volatile chemical products (VCPs)—including pesticides, coatings, printing inks, adhesives, cleaning agents, and personal care products now constitutes half of fossil fuel VOC emissions in industrialized cities."

Okay... That is not what the linked article says at all.