Gersten complexes in Quillen's and Milnor's K-theories

Consider a good enough scheme $X$ (e.g. an algebraic variety over a field). Let $X_i$ be the set of points of dimension $i$ in $X$. Then we have the Gersten complex in Quillen's K-theory:

$$\oplus_{x\in X_{i+1}}K_{n+1}(\kappa(x))\to \oplus_{x\in X_i}K_n(\kappa(x))\to \oplus_{x\in X_{i-1}}K_{n-1}(\kappa(x))$$

Kato proved that there is a Gersten complex of the same type using Milnor's K-groups instead. As there are natural maps from Milnor K-groups to Quillen K-groups, I would like to know whether there is a compatibility between (the differential maps in) these two complexes.

For the differentials from $K_2$ to $K_1$ or from $K_1$ to $K_0$, it is well-known that the two complexes give the same thing (up to a sign). But I'm not sure if someone has checked this for higher $K$-groups. Could anyone give some hints or references?

That paper contains a definition of "cycle modules" and constructions of Gersten complexes for these. Milnor K-theory is almost by definition a cycle module, and Quillen K-theory is a cycle module by Remark 1.12. Remark 5.4 in Rost's paper states that the multiplication morphisms yield a morphism of cycle modules, and by the results and constructions in the paper, a morphism of cycle modules induces a morphism of corresponding Gersten complexes. (Note that Rost references the 1987 paper of Merkurjev and Suslin for an identification of the differentials for $K_3$.)