TITLE: References for $K_{4k}(\mathbb{Z})$
QUESTION [23 upvotes]: Weibel's "Algebraic K-theory of rings of integers in local and global fields" says $K_{4k}(\mathbb{Z})$ are known to have odd order, with no prime factors less than $10^7$, but are conjectured to be zero.
Since the paper is from 2004 I was wondering if anything new is known about those groups.
I'm interested on any new research on them.

REPLY [5 votes]: $K_8(\mathbb{Z})=0$
A proof of the result announced by Elbaz-Vincent and mentioned above can now be found in:

Mathieu Dutour-Sikirić, Philippe Elbaz-Vincent, Alexander Kupers, and Jacques Martinet, Voronoi complexes in higher dimensions, cohomology of $GL_N(\mathbb{Z})$ for $N\geq8$ and the triviality of $K_8(\mathbb{Z})$, arXiv:1910.11598v1 (submitted on October 25, 2019)

A shorter proof, which (unlike the one above) does not require additional computer calculations, is presented in:

Alexander Kupers, A short proof that $K_8(\mathbb{Z})=0$, http://people.math.harvard.edu/~kupers/files/k8zshorter.pdf (version of October 21, 2019)

The very last remark in Kupers’s short note addresses the case of $K_{12}(\mathbb{Z})$:

Remark 2.6. If Conjecture 2 of [CFP14] were true with coefficients in $\mathbb{Z}[1/((n+1)!)]$ instead of $\mathbb{Q}$, it could be used to prove that $K_{12}(\mathbb{Z})=0$ in a similar manner.
[CFP14] Thomas Church, Benson Farb, and Andrew Putman, A stability conjecture for the unstable cohomology of $SL_n(\mathbb{Z})$, mapping class groups, and $Aut(F_n)$, Algebraic topology: applications and new directions, 55–70, Contemp. Math., vol. 620, Amer. Math. Soc., 2014, MR3290086