Word problem and quotient group

Let $G$ be given by the group presentation $G = \langle a,b \mid S \rangle$, where $S = \{aa,bbb\}$. The formal definition of $G$ is:

Let $F$ be the free group on $\{a,b\}$. Let $H$ be the conjugate closure of $S$, i.e. $H$ is the subgroup of $F$ generated by $\{fsf^{-1} : f \in F, s \in S\}$. Then we can define the quotient group $G = F/H$.

Let $x,y \in F$ be words consisting of only letters $a, b$. Using the formal definition of $G$ above, how can we show that if $x=y$ in $G$ (i.e. $xH = yH$), then there is a finite sequence of operations changing $x$ to $y$ where in each operation we either add/remove an $aa$ or add/remove a $bbb$ to $x$?

Thoughts: We can write $y^{-1}x = \text{reduce}(f_1 s_1 f_1^{-1} f_2 s_2 f_2^{-1} \dots f_k s_k f_k^{-1})$, where $f_i \in F$, and $s_j \in S' = \{aa,bbb,a^{-1}a^{-1},b^{-1}b^{-1}b^{-1}\}$. Here reduce($w$) is the reduction of $w$ in the free group by cancellation of consecutive $a,a^{-1}$ or $b,b^{-1}$. But I'm not sure how to proceed from here.

Edit: Can we also show this holds for arbitrary $S$? For example, $S = \{aa,bbb,ababab\}$?

from Hot Weekly Questions - Mathematics Stack Exchange