The Julia-set conjecture for the real ${\mathcal S}_{5}$-invariant ${\mathbf{RP}^{3}}$ of $h_{11}$

Let $h_{11}$ be the map under discussion, and let ${\mathcal R}$ be the ${\mathcal S}_{5}$-symmetric real projective subspace ${\mathbf{RP}^{3}}$ preserved by $h_{11}$. The Julia set is the set of points where the iterates do not form a normal family. Julia-set conjecture. The ${\mathcal S}_{5}$-invariant ${\mathbf{RP}^{3}}$ is non-attracting, possibly repelling, and therefore belongs to the Julia set of $h_{11}$. This claim is motivated by iteration experiments showing attraction only to the ten chaotically attracting ${\mathbf{RP}^{1}}$ intersections with the relevant $10$-lines; the supplied text gives no resolution.