Does linear memory effect encode in the gravitational waves?

I read from a few papers that the gravitational memory effect has linear and nonlinear parts: $$\Delta h^{TT}=\Delta h^{TT}_{linear}+\Delta h^{TT}_{nonlinear}$$ and the nonlinear part is encoded in the GW.

But the linear part will also cause permanent change in metric at null infinity, so the information of this effect should be also carried by some media, which to me is nothing but GW.

But when people calculate the waveform of memory (like arXiv:1807.00990), they only consider the nonlinear part, I don't understand why, and I can't find any explanation about this.

So if I misunderstand something about the memory effect here, can someone kindly help me figure it out? or explain how exactly the permanent change happens?