Let $G=A\ast \mathbb{Z}$ be the free product of a group $A$ and the cyclic group $\mathbb{Z}$ and suppose $K$ is a subgroup of $G$. By Kurosh Subgroup Theorem we know that $K=F\ast (\ast_{i\in I}(K\cap A^{u_i}))$, where $F$ is free group and $u_i$ are some representatives of double cosets $KxA$ in $G$.

Now suppose further that $A$ has ACC on normal subgroups and $K$ is normal.

Is it true that $K$ is finitely generated?

(This will be true if we can show that $|I|$ and $\operatorname{rank}\ F$ are finite.)