Statistics of $K$-groups modulo $p$ for the ring of integers of a varying quadratic number field

For each odd prime $p$, we conjecture the distribution of the $p$-torsion subgroup of $K_{2n}({\mathcal{O}}_F)$ as $F$ ranges over real quadratic fields, or over imaginary quadratic fields. We then prove that the average size of the $3$-torsion subgroup of $K_{2n}({\mathcal{O}}_F)$ is as predicted by this conjecture.