One of the properties of the Chow group of a scheme is that it is homotopy invariant, that is to say that for any scheme $X$ over some field $k$, we have $$\text{Ch}^n(X\times \mathbb{A}^1)\cong \text{Ch}^n(X).$$ The Chow group is the quotient of the free group of algebraic cylces of some co-dimension up to the equivalence class of rational equivalence. Amongst the class of adequate equivalence relations of algebraic cylces, this is one of the stronest (https://en.wikipedia.org/wiki/Adequate_equivalence_relation) (by which I mean it is the strongest mentioned example in the literature I've read).
Is the same statement true for weaker equivalence classes? Say at the "other end" of the spectrum, for instance numerical equivalence, i.e. do we have $$Z^n(X\times \mathbb{A}^1)/Z^n_{num}(X\times \mathbb{A}^1)\cong Z^n(X)/Z^n_{num}(X)?$$
I haven't found the statement or a counterexample in Fulton's book and other places, so I don't know what to belive.