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2 finite rank

Let $F$ be a free group of finite rank $r>1$ and let $c_1,c_2,...$ be a full set of basic commutators for $F$. Upon completing the $k$-th step of the collection process for an arbitrary $g\in F$, one obtains a representation

$g=c_1^{n_1}c_2^{n_2}\ldots c_{t}^{n_t}g_k$,

where

$c_1,\ldots,c_t$ are all basic commutators of weight less than $k$ ($t$ depending on $k$),

$n_1,\ldots,n_t$ are integers,

$g_k\in \gamma_k$, the $k$-th term of the lower central series for $F$.

Question. Is there always $g\in G$ such that the series $g_1,g_2,\ldots$ never terminates (i.e. has nonidentity terms $g_i$ for arbitrarily large $i$)?

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# Is the collection process for commutators potentially infinite?

Let $F$ be a free group of rank $r>1$ and let $c_1,c_2,...$ be a full set of basic commutators for $F$. Upon completing the $k$-th step of the collection process for an arbitrary $g\in F$, one obtains a representation

$g=c_1^{n_1}c_2^{n_2}\ldots c_{t}^{n_t}g_k$,

where

$c_1,\ldots,c_t$ are all basic commutators of weight less than $k$ ($t$ depending on $k$),

$n_1,\ldots,n_t$ are integers,

$g_k\in \gamma_k$, the $k$-th term of the lower central series for $F$.

Question. Is there always $g\in G$ such that the series $g_1,g_2,\ldots$ never terminates (i.e. has nonidentity terms $g_i$ for arbitrarily large $i$)?