Here I am asking for an analogue of Generating infinite index subgroups of a free group

Let $F$ be a nonabelian finitely generated free group, let $H \leq F$ be a finitely generated subgroup of infinite index, and let $x \in F$. Must there be some $y \in F$ such that $[F : \langle H, yxy^{-1}\rangle] =\infty$ ?