Quillen's freeness conjecture for special linear groups over rings of integers

Let $\ell$ be a prime number. Let $K$ be a number field with $\zeta_\ell\in K$, and let $S$ be a finite set of places containing the infinite places and the places over $\ell$. The natural inclusion

$$\mathcal{O}_{K,S}\hookrightarrow \mathbb{C}$$

induces a module structure on $\operatorname{H}^*(\operatorname{SL}_n(\mathcal{O}_{K,S});\thinspace \mathbb{F}_\ell)$ over the continuous cohomology ring $\operatorname{H}^*_{\operatorname{cts}}(\operatorname{SL}_n(\mathbb{C});\thinspace \mathbb{F}_\ell)$, and Quillen's conjecture. This module is free. The conjecture predicts a uniform freeness property for the mod-$\ell$ cohomology of arithmetic special linear groups over the cohomology of the corresponding complex Lie group; the paper verifies it in the rank-$2$ imaginary quadratic case considered there.