Here is the question:

if $X$ is a separated, finite type scheme over a perfect field (but not necassarily smooth) is the map $KH_n(X) \to \prod_{x \in X^{(0)}} KH_n(k(x))$ injective?

If $X$ is smooth, this is known for the Zariski sheaf associated to $KH_n$. I am wondering if anyone knows off the top of their head how essential smoothness is to the proof.

Edit: I originally mistakenly stated that $KH$ is a presheaf with transfers; its not (see page 105 of "Cohomological theory of presheaves" - Voeovdsky and "Triangulated categories of motives over a field" 3.1.11) however, its Zariski sheafification does have transfers on smooth schemes after work by Deglise.

Edit 2: (after the comment by Cisinski): I originally was missing "for the Zariski sheaf associated to $KH_n$".