Odabaş's arithmetic exceptionality conjecture for Lattès maps

Let $k$ be a positive integer, let $E/\mathbb{Q}$ be an elliptic curve, and let $L_k$ be the $k$-th Lattès map attached to $E$. A rational function is arithmetically exceptional if it induces a permutation of $\mathbb{P}^1(\mathbb{F}_p)$ for infinitely many primes $p$. Odabaş's conjecture. The map $L_k$ is arithmetically exceptional if and only if $E$ possesses no $k$-torsion point whose $x$-coordinate is rational. This conjecture connects permutation behavior of Lattès maps over finite fields with the rational torsion structure of elliptic curves. The paper proves it for elliptic curves with complex multiplication by an imaginary quadratic field other than $\mathbb{Q}(\sqrt{-11})$, and shows that it fails for curves with complex multiplication by $\mathbb{Q}(\sqrt{-11})$ when $6\mid k$; partial results are also obtained for non-CM curves.