Conjecture on symmetric invariant sets at the values $\varepsilon_n$

Let $\mathcal{S}$ be the nontrivial symmetric invariant set for the four-site system, and define

$$\varepsilon_n=1-\frac{\sqrt[2^n]{2}}{2},\qquad n>1.$$

Symmetric invariant-set conjecture. At every $\varepsilon_n$, new symmetric invariant sets appear; these sets are contained in $\mathcal{S}$ and are symmetric. The conjecture is motivated by simulations and by the increasing number of mixing components in the associated one-dimensional map, but no proof is supplied.