I am trying to solve an exercise from Clara Loh's Geometric Group Theory: An introduction. The problem uses the exponential growth rate of a finitely generated group $G$ with generating set $S$. The exponential growth rate is then defined as the limit $$ \rho_{G,S} = \inf_{r\in \mathbb{N }_{>0}} (\beta_{G,S}(r))^{1/r}, $$ where $\beta_{G,S}$ is the growth function of $G$.
The question is then the following: Given a free group F of rank 2, freely generated by $\{a,b\}$ and consider the set $S:= \{a,b,aba^{-1},a^2\} \subset F $. Show that $$\rho_{F,\{a,b\}} \neq \rho_{F,S}. $$
I know that $\rho_{F,\{a,b\}} = 3$, since for general free groups of rank $n$ and freely generating set $T$ we have $$\beta_{G,T} (r) = 1 + \frac{n}{n-1} ( (2n-1)^r -1).$$ Moreover, I know that we can always find a subset of $S$ (in this case $\{a,b\}$) which will freely generating a free subgroup of $F$ of rank 2. Hence $\rho_{F,S} \geq \rho_{F,\{a,b\}} = 3$.
So I am trying to prove that this inequality is strict. For this I tried to understand $\beta_{F,S}(r)$ by drawing a Calay-graph up to $r=2$. For larger $r$ this gets a bit messy. But what I do observe from drawing this, is the fact that $\beta_{F,S}(r) > \beta_{F,\{a,b\}}(r)$. But taking the infimum does not guarantee strictness of this inequality.
I am somehow hoping that it is possible to show $\beta_{F,S}(r)$ grows by an additional factor of $3^r$, but I am far from prooving anything like this.
Some hints on how to tackle this problem are very welcome. Thank you.