Suppose, $G = \mathbb{Z} \ast H$, where $H$ is an arbitrary group. Suppose, $g \in G$ and $g \notin \langle\langle H \rangle \rangle $. Is $\langle\langle g \rangle \rangle \cap H$ always trivial? ($\ast$ stands for free product, and $\langle \langle \dots \rangle \rangle$ stands for normal closure)

Yesterday, I have asked this question on math.stackexchange.com and was advised to re-ask it there: Is the intersection of two subgroups, defined below, always trivial? v2.0

Any help will be appreciated.