Topological connectivity of random permutation complexes

Let $\mathbb{S}_n$ denote the symmetric group on $[n]=\{1,\ldots,n\}$ with the uniform probability measure. For a permutation $\pi \in \mathbb{S}_n$ let $X_{\pi}$ denote the simplicial complex on the vertex set $[n]$ whose simplices are all $\{i_0,\ldots, i_m\} \subset [n]$ such that $i_0<\cdots<i_m$ and $\pi(i_0)<\cdots < \pi(i_m)$. For $r \geq 0$ let $p_r(n)$ denote the probability that $X_{\pi}$ is not topologically $r$-connected for $\pi \in \mathbb{S}_n$. It is shown that for fixed $r \geq 0$ there exist constants $0<C_r, C_r' < \infty$ such that $$C_r \frac{(\log n)^r}{n} \leq p_r(n) \leq C_r' \frac{(\log n)^{2r}}{n}.$$