Recall that for a DVR A with fraction field F and residue field k, there is a "localization" fiber sequence in algebraic K-theory,
$$K(k) \rightarrow K(A) \rightarrow K(F).$$
In Remark 5.17 of his "Higher Algebraic K-theory: I" paper, Quillen gives an explicit description of the corresponding boundary map $\partial:\Omega K(F) \rightarrow K(k)$, saying the proof will be in a later paper. My question is, has a proof appeared in the literature? I'd also be happy with proofs in the literature of any similar descriptions, e.g. involving the S-dot construction.
Thank you for reading!