Homology conjecture for flag complexes of complete digraphs

Let $G_n$ be a complete digraph on $n$ nodes, and let $\dFl(G_n)$ denote its flag complex. Write $H_k(\dFl(G_n))$ for the $k$-th homology group, and let $!n$ be the number of derangements of $\\{1,\dots,n\\}$. Homology conjecture for complete digraphs. If $n>1$, then

$$\dim H_k(\dFl(G_n)) = \begin{cases} 1 & \text{if } k=0,\\ \\ !n & \text{if } k=n-1,\\ \\ 0 & \text{otherwise}. \end{cases}$$

Although $\dFl(G_n)$ is not contractible for $n>1$, this conjecture predicts that its homology is concentrated in dimensions $0$ and $n-1$. It has been verified computationally for $n\leq 8$ using Flagser, but no general proof is given here.