On a base change conjecture for higher zero-cycles

We show the surjectivity of a specialisation map on higher $(0,1)$-cycles for a smooth projective scheme over an excellent henselian discrete valuation ring. This gives evidence for a conjecture stated in an article of Kerz, Esnault and Wittenberg saying that base change holds for such schemes in general for motivic cohomology in degrees $(i,d)$ for fixed $d$ being the relative dimension over the base. Furthermore, the specialisation map we study is related to a finiteness conjecture for the $n$-torsion of $CH_0(X)$, where $X$ is a variety over a $p$-adic field.