Dual Gottschalk conjecture for post-surjective cellular automata

Let $G$ be a group and let $A$ be a finite set. Let $\tau\colon A^G\to A^G$ be a cellular automaton. It is post-surjective if, for every $x,y\in A^G$ such that $y$ is asymptotic to $\tau(x)$, there exists $z\in A^G$ asymptotic to $x$ with $\tau(z)=y$. It is pre-injective if $\tau(c)=\tau(d)$ implies $c=d$ whenever $c,d\in A^G$ are asymptotic. Dual Gottschalk conjecture. If $\tau\colon A^G\to A^G$ is post-surjective, then $\tau$ is pre-injective. This is the dual version of Gottschalk's conjecture, introduced by Capobianco, Kari, and Taati. Its resolution is not indicated in the supplied text.