Lee–Szczarba conjecture on top-dimensional cohomology of congruence subgroups

Let $\Gamma_n(p)$ be the level-$p$ principal congruence subgroup of $\operatorname{SL}_n(\mathbb{Z})$, and let ${\mathcal{T}}_n(\mathbb{Q})$ be the Tits building of $\operatorname{SL}_n(\mathbb{Q})$. Write

$$\operatorname{St}_n(\mathbb{Q})=\widetilde{\operatorname{H}}_{n-2}({\mathcal{T}}_n(\mathbb{Q})).$$

The quotient map from ${\mathcal{T}}_n(\mathbb{Q})$ to ${\mathcal{T}}_n(\mathbb{Q})/\Gamma_n(p)$ induces a map on reduced homology.

Lee–Szczarba conjecture. For a prime $p$ and $n\geq 2$, the induced map

$$(\operatorname{St}_n(\mathbb{Q}))_{\Gamma_n(p)}\longrightarrow \widetilde{\operatorname{H}}_{n-2}({\mathcal{T}}_n(\mathbb{Q})/\Gamma_n(p))$$

is an isomorphism. Equivalently, the top-dimensional cohomology group $\operatorname{H}^{\binom{n}{2}}(\Gamma_n(p))$ is isomorphic to $\widetilde{\operatorname{H}}_{n-2}({\mathcal{T}}_n(\mathbb{Q})/\Gamma_n(p))$.

The conjecture identifies all top-dimensional cohomology with the coinvariants of the rational Steinberg module. The source states that it was proved for $p=3$, while the paper explains that larger primes require additional cohomology beyond the Steinberg module over $\mathbb{F}_p$; the supplied text does not establish a complete resolution for all primes.