In the appendix of this paper. It is proved that Bass’ conjecture for $ K_n$ implies the rational Beilinson-Soulé conjecture for $ K_n$ . Then at the end the author claims that the same method can be applied to prove that the Bass’ conjecture implies the Parshin’s conjecture but I can’t figure it out. The general idea behind the Beilinson-Soulé conjecture is that he proves the Theorem A.1 for fields then uses the Quillen spectral sequence to prove it for the general regular scheme $ X$ which I can’t see how this idea can be used for the Parshin conjecture. I wonder whether this proof is written anywhere with more details. I’d appreciate if anyone of the experts in the field can explain the sketch of the proof for the Parshin’s conjecture or answer this question:

Is the finite generation of $ K_0$ required to imply the Parshin’s conjecture or the finite generation of $ K_n$ for $ n\geq 1$ is enough?