Free-product closure conjecture for dual surjunctive groups

Let $G$ be a group, and let $\mathbb{Z}*G$ denote its free product with the infinite cyclic group. A group is dual surjunctive if every post-surjective cellular automaton over it is pre-injective. Free-product closure conjecture. If $G$ is (dual) surjunctive, then $\mathbb{Z}*G$ is (dual) surjunctive as well. Such a closure property would enlarge the classes of groups known to be surjunctive and dual surjunctive; the source presents it as a conjecture.