In a paper by S. Landsburg the (higher) Chow groups of a pair $(X,Y)$ are defined when $Y$ is a smooth closed subvariety of a smooth variety $X$ as follows.
We consider the sub-complex $z^{*}(X;.)_{Y}$ of Bloch's complex $z^{*}(X;.)$ which is generated by subvarieties of $X\times\Delta^n$ that meet $Y\times F$ properly where $F$ is any face of $\Delta^n$. The author then defines a restriction map $r:z^{*}(X;.)_Y\to z^{*}(Y;.)$ (presumably this is a variant of the map defined a la Serre's Algebre Locale Multiplicités where the multiplicity attached to an irreducible component of the intersection is given by the alternating sum of the lengths of the $\mathrm{Tor}$'s). After this $z^{*}(X,Y;.)$ is defined as $\mathrm{Cone}(r)[-1]$. The Chow groups $\mathrm{CH}^{*}(X,Y;.)$ are defined as suitable (co-)homologies of this complex.
However, it seems (e.g. see the review of the paper as well as footnote 1 in the paper) this paper has some unresolved issues due to its dependence on some unpublished work and conjectures of Bloch. Hence, one would like to know if the above definition "works" in conjunction with the expected long-exact sequence involving the Chow groups of $X$ and $Y$. (In this form the question being asked here seems to be a special case of a question asked earlier.) For example, is the above geometric description of the Chow Groups of the pair valid when one uses the definition of the Chow groups in terms of motives?
A natural way to get this to work, would be to prove that the natural inclusion $z^{*}(X,.)_Y\to z^{*}(X,.)$ is a quasi-isomorphism. Looking for possible proofs of this, one finds the book by Mazza, Weibel and Voevodsky(MWV) in which proposition 17.6 gives a similar result (via a reference to theorem I.II.3.15.4 in the book by Levine). However, both these references restrict the assertion to the case when $X$ is affine. (Our $X$ and $Y$ are interchanged in [MWV]!)
With this background, here are the questions:
Is proposition 17.6 of [MWV] valid in this restricted form without the assumption that $X$ is affine? For example, is the natural inclusion a quasi-isomorphism when $X$ is quasi-projective?
If the answer is "No!" to (1), then is it only that the given proof fails, or is there a known counter-example to this proposition?
A simpler version of this question is when $Y$ is a smooth divisor in $X$ and so one does not need to worry about higher $\mathrm{Tor}$'s in the definition. Is Landsburg's description valid in that case?
A weaker question than all of the above is whether there is a "simple" geometric description of $\mathrm{CH}^{*}(X,Y;.)$ for a smooth divisor $Y$ in a smooth variety $X$. (Here we use the definition of this Chow group in terms of motivic cohomology.)
