Formal solutions and the first?order theory of acylindrically hyperbolic groups

We generalise Merzlyakov's theorem about the first-order theory of non-abelian free groups to all acylindrically hyperbolic groups. As a corollary, we deduce that if $G$ is an group and $E(G)$ denotes unique maximal finite normal subgroup $G$, then HNN extension $G\dot{\ast}_{E(G)}$, which simply product $G\ast\mathbb{Z}$ when trivial, have same $\forall\exists$-theory. consequence, prove following conjecture, formulated by Casals-Ruiz, Garreta de la Nuez Gonz\'alez: trivial positive theory. In particular, one recovers result proved Bestvina, Bromberg Fujiwara, stating that, with only obvious exceptions, verbal subgroups infinite width.