  Let $\Gamma_n(p)$ be the level-$p$ principal congruence subgroup of
$\text{SL}_n(\mathbb{Z})$. Borel-Serre proved that the cohomology of
$\Gamma_n(p)$ vanishes above degree $\binom{n}{2}$. We study the cohomology in
this top degree $\binom{n}{2}$. Let $\mathcal{T}_n(\mathbb{Q})$ denote the Tits
building of $\text{SL}_n(\mathbb{Q})$. Lee-Szczarba conjectured that
$H^{\binom{n}{2}}(\Gamma_n(p))$ is isomorphic to
$\widetilde{H}_{n-2}(\mathcal{T}_n(\mathbb{Q})/\Gamma_n(p))$ and proved that
this holds for $p=3$. We partially prove and partially disprove this conjecture
by showing that a natural map $H^{\binom{n}{2}}(\Gamma_n(p)) \rightarrow
\widetilde{H}_{n-2}(\mathcal{T}_n(\mathbb{Q})/\Gamma_n(p))$ is always
surjective, but is only injective for $p \leq 5$. In particular, we completely
calculate $H^{\binom{n}{2}}(\Gamma_n(5))$ and improve known lower bounds for
the ranks of $H^{\binom{n}{2}}(\Gamma_n(p))$ for $p \geq 5$.
