Let $X$ be a smooth quasiprojective algebraic variety over a field $k$. Then the $K$-groups $K_m(X)$ are defined, and there are two standard filtrations on them: the "codimension filtration" given by $$\operatorname{Fil}^p_{\mathrm{cod}} K_m(X) :=\bigcup\nolimits_Y \text{Ker}(K_m(X)\to K_m(X-Y))$$ where the limit is taken over all $Y$ of codimension $\ge p$; and the "$\gamma$-filtration" $\operatorname{Fil}^\bullet_\gamma$ defined by Quillen. I gather that $\operatorname{Fil}^p_\gamma \subset \operatorname{Fil}^{p-m}_{\mathrm{cod}}$, and that for $m = 0$, this becomes an isomorphism after tensoring with $\mathbb{Q}$.

Are there other cases (maybe under additional assumptions on $k$ or on $X$) where the two filtrations agree rationally? Grayson's article in "Handbook of $K$-theory" suggests that they should be different when $X = \mathrm{Spec}(k)$ and $k$ has large cohomological dimension, but I confess that I don't understand the reasoning behind this.

The two filtrations both come from spectral sequences: the codimension filtration from the Brown--Gersten--Quillen sequence $$E_2^{pq} = H^p(X, \mathscr{K}_{-q}),$$ where $\mathscr{K}_q$ denote the $K$-theory sheaves on $X$; and the $\gamma$-filtration (if I've undstood correctly) comes from the Friedlander--Suslin motivic spectral sequence $$E_2^{pq} = \operatorname{CH}^{-q}(X,-p-q) = H^{p-q}_{\mathrm{mot}}(X, \mathbb{Z}(-q)).$$

Both converge to $K_{-p-q}(X)$ (sorry about the convoluted indexing!). Landsburg ("Relative Chow groups", Illinois J Math (35), 1991) has constructed maps $\operatorname{CH}^{-q}(X,-p-q) \to H^p(X, \mathscr{K}_{-q})$ between the $E_2$ terms of the spectral sequences, which he shows are isomorphisms if $p + q = 0$ or $-1$.

Are Landsburg's maps compatible with the differentials in the two spectral sequences, and are there additional hypotheses which would force them to be isomorphisms (maybe up to torsion) in all degrees?

(Apologies if this is a naive question; I'm only just beginning to learn all of this stuff.)