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?

Edit: Some thoughts: I believe the proof should be some sort of induction and using the Quillen spectral sequence we should be able to reduce the problem to the function field of projective varieties. The inductive step should be something like showing the algebraic $K$-groups of the function field $F$ of a smooth projective variety over a finite field (of char $p$) is uniquely $p$-divisible after multiplying it with some finite number $N$. Since the Milnor $K$-theory of fields of finite char coincides with the algebraic $K$-theory up to uniquely divisible groups this reduces to showing that $K^M_i(F)$ is uniquely $p$-divisible after multiplying it with some finite number $N$. Is this something known?


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I think the claim in the original paper linked above is not correct. It is either totally wrong or it is meant to be something else (like etale motivic version of Bass conjecture) which again I'm not sure whether the same argument given for the Beilison-Soule works or not (I cannot see how it works.). As a reference you can take a look at here page 15 the paragraph after Theorem 8.1.


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