I don’t know. I just googled all of these terms as I am … old. Origin of “no cap” comes from “high capping” (rap lyrics). https://amp.knowyourmeme.com/memes/no-cap-cap

I think it refers to a cap (hat), because culturally when you take off your cap it means you are being truthful. Similar to the term “hat in hand”

I looked into it more and seems like the origins of cap meaning to lie is unclear. So looks like I was off base