According to this article [0], you're right:

> This property of the strong nuclear force is known as asymptotic freedom, and the particles that mediate this force are known as gluons. Somehow, the energy binding the proton together, the other 99.8% of the proton's mass, comes from these gluons.

[0] https://www.forbes.com/sites/startswithabang/2016/08/03/wher...

Yep, that’s correct. I misspoke (should have said “not due to rest mass”). You can think of this by analogy with the Coulomb Hamiltonian—it is the sum of a kinetic term and an electromagnetic potential term. With the nucleus, it’s similar except the potential term corresponds to the strong force and is even much higher than the kinetic term as you say.