Lemma: A domain for both free orderings

The space $D_{\mathrm{cmp}}$ is stable under $\sigma$ and $\gamma$. Both orderings of every pair represented there are declared. In particular, $D_{\mathrm{first}}\subset D_{\mathrm{cmp}}$.

Proof. Flip stability follows directly from ${\sigma }^2=1$. Since $\sigma \gamma (D_{\mathrm{decl}})=D_{\mathrm{decl}}$, one has $\gamma (D_{\mathrm{decl}})=\sigma (D_{\mathrm{decl}})$. Therefore

The first-stage sum contains both orderings, and $\mathcal{B},\mathcal{W}$ are conjugation stable.