at.algebraic topology – Commutator length of the fundamental group of some grope

A popular way to describe a grope as the direct limit $L$ of a nested sequence of compact 2-dimensional polyhedra
$L_0 to L_1 to L_2 to cdots$
obtained as follows. Take $L_0$ as some $S_g$, an oriented compact surface of positive
genus $g$ from which an open disk has been deleted. To form $L_{n+1}$ from $L_n$, for each
loop $a$ in $L_n$ that generates the first homology group $H_1(L_n)$, attach to $L_n$ some $S_{g_a}$ by identifying
the boundary of $S_{g_a}$ with the loop $a$. Since the fundamental group of $S_g$ punctured
is a free group on $2g$ generators, this procedure embeds each $pi_1(L_n)$, and thus
each finitely generated subgroup of $pi_1(L)$, as a subgroup of a free group, with each
generator $a$ of $pi_1(L_n)$ becoming a product of $g_a$ commutators in $pi_1(L_{n+1})$. Hence
$pi_1(L)$ is a countable, perfect, locally free group. See What is a grope? for some motivation.

Here we build a grope $A$ in stages. At each stage put in disks with 3 handles on all the various handle curves at that stage. In other words, grope $A$ is built using disks with 3 handles uniformly throughout.

Question. Is it possible that there is a generating set $G$ of $pi_1(A)$ such
that each element of $G$ has commutator length bounded by 2?