Let $F$ be a free group of finite rank. Does $F$ satisfy the ascending chain condition on normal subgroups each of which is a normal closure of one element?

In other words, can there exist elements $r_1,r_2,\dots\in F$ such that $\langle r_i^F \rangle \subsetneqq \langle r_{i+1}^F \rangle$, for all $i \in \mathbb{N}$?

I'm sure the answer should be known, but I have not yet been able to find a reference for it.

Let me add some background.

- B.H. Neumann [An essay on free products of groups with amalgamations. Philos. Trans. Roy. Soc. London Ser. A 246 (1954), 503–554.] proved that in the free group of rank $3$, $F_3$, there is an infinite strictly ascending sequence of normal subgroups $N_1 \subsetneqq N_2 \subsetneqq \dots$, such that each $N_i$ is the normal closure of
**two**elements in $F_3$. Moreover, there is an automorphism $\alpha \in Aut(F_3)$ such that $N_{i+1}=\alpha(N_i)$, $i \in \mathbb N$. - A.M. Brunner [On a class of one-relator groups. Canadian J. Math. 32 (1980), no. 2, 414–420.] constructed a sequence of (pairwise non-isomorphic and Hopfian) one-relator groups $G_0,G_1,\dots$, such that for all $i<j$, $G_j$ is a proper quotient of $G_{i}$ by the normal closure of a single element. (Unfortunately this does not answer my question since the group $G_0$ is not free.)