Gersten's arithmetic DVR conjecture

Let $(R,pR,k)$ be a discrete valuation ring essentially smooth over $\mathbb{Z}$, with residue field $k$. Gersten's Arithmetic DVR Conjecture. For every $n \geq 0$, the transfer map

$$K_n(k) \longrightarrow K_n(R)$$

is zero. This conjecture is presented as the reduction needed to study Gersten's conjecture for unramified regular local rings of mixed characteristic. The source gives no resolution status for this transfer-map assertion.