Conjecture on the second asymmetric invariant set $\mathcal{A}_2$

Let $P_3$ be a polyhedron in $\mathbb{T}^3$, let $S_0,\ldots,S_6$ be the specified symmetries of the map $G_{\varepsilon,3}$, and define

$$\mathcal{A}_2=P_3\cup S_1(P_3)\cup S_2(P_3)\cup S_4(P_3)\cup S_2S_1(P_3)\cup S_1S_2(P_3).$$

Second asymmetric-set conjecture. There exists such a polyhedron $P_3$ for which $\mathcal{A}_2$ is symmetric with respect to $S_1,S_2$ and $S_4$, asymmetric with respect to $S_0,S_3,S_5$ and $S_6$, and invariant whenever

$$0.438\approx\frac{5-\sqrt{17}}{2}=\varepsilon^{**}\leq\varepsilon.$$

This gives the paper's explicit proposed value for the second critical coupling at which further asymmetric invariant sets emerge; the claim is based on calculations and remains conjectural.