Here are some lectures by Chuck Weibel. Early on, they discuss Milnor Conjecture and Bloch-Kato, and they should go through the proof. My understanding is that there were a bunch of people involved in the proof, though a few were a bit reticent to actually write up their parts of it, and so Weibel drew the short straw and is the one writing it up.

Here are some lectures by Charles Weibel. Early on, they discuss Milnor Conjecture and Bloch-Kato, and they should go through the proof. My understanding is that there were a bunch of people involved in the proof, though a few were a bit reticent to actually write up their parts of it, and so Weibel drew the short straw and is the one writing it up.

EDIT: Adding a bit, here's Weibel's 2006 page where he notes the status as of then, and to make sure that this is roughly self-contained, here's the statement:

For an odd prime $\ell$, and a field $k$ containing $1/\ell$, the Milnor K-theory $K^M_n(k)/\ell$ is isomorphic to the étale cohomology $H^n_{ét}(k,μ_\ell^n)$ of the field $k$ with coefficients in the twists of $μ_\ell$.