Csorba's Hom-complex model for Stiefel manifolds

For a graph $G$, let $\text{\tt Hom}(G,H)$ denote its Hom-complex. Let $V_2(\mathbb R^{n-1})$ be the Stiefel manifold of orthonormal $2$-frames in $\mathbb R^{n-1}$. Csorba's conjecture. For every $n\geq 1$, the complex $\text{\tt Hom}(C_5,K_n)$ is homeomorphic to $V_2(\mathbb R^{n-1})$. The cases $n=1,2$ are tautological, and the source reports verification through $n=4$; no general proof is supplied.