I have asked this question in MSE but have not got any satisfactory answer, so I am asking it here. Any idea on how to approach this problem will be highly appreciated.

Consider the Hawaiian earring. Suppose $f_n$'s are the loops representing the circles of radius $1/n$ centered at $(1/n, 0)$ for $n=1,2\cdots$. Suppose the following holds in the fundamental group

$$\langle [f_1, f_2][f_3, f_4]\cdots\rangle = \langle [g_1, g_2]\cdots [g_{2k-1},g_{2k}]\rangle $$ for some loops $g_1,g_2,\ldots , g_{2k}$, where $\langle\cdot\rangle$ denotes the homotopy class and $[a,b]=aba^{-1}b^{-1}$ denotes the commutator. Then can we say that $$\langle [f_1, f_2]\cdots [f_{2n-1}, f_{2n}]\rangle = \langle [g_1, g_2]\cdots [g_{2k-1},g_{2k}]\rangle$$ for all sufficiently large $n$, or for some $n>k$ ?

I had this question while I was reading the paper here (see page 76, last paragraph), where the above has been mentioned (in some other equivalent form) without proof.