Uniqueness conjecture for subgroups isomorphic to the alternating group in the real space Cremona group

Let $\mathrm{Cr}_3(\mathbb{R})$ denote the Cremona group of birational self-maps of real projective 3-space, and let $\mathfrak{A}_6$ denote the alternating group on six letters. Uniqueness conjecture. Up to conjugation, $\mathrm{Cr}_3(\mathbb{R})$ contains a unique subgroup isomorphic to $\mathfrak{A}_6$. The paper has established that $\mathfrak{A}_6$ occurs, via the action on the real-rational Segre cubic threefold, but the uniqueness of its conjugacy class is left as a conjectural statement.