Relative $K$-theory via 0-cycles in finite characteristic

Let $R$ be a regular semi-local ring, essentially of finite type over an infinite perfect field of characteristic $p\ge 3$. We show that the cycle class map with modulus from an earlier work of the authors induces a pro-isomorphism between the additive higher Chow groups of relative 0-cycles and the relative $K$-theory of truncated polynomial rings over $R$. This settles the problem of equating 0-cycles with modulus and relative $K$-theory of such rings via the cycle class map.