The realization graph of every degree sequence has a Hamilton path

For a degree sequence $d$, the realization graph $\mathcal{G_F}(d)$ is the graph whose vertices are the labeled realizations of $d$, two of which are adjacent if they differ by a single $2$-switch. We prove that for every degree sequence $d$ and every realization $S$ of $d$, the graph $\mathcal{G_F}