Let $G$ be a free group generated by a set $S$. For $g\in G$, let $l(g)$ be the length of $g$ with respect to $S$.

Now for $a\in G$ and $g_1,\dotsc,g_n\in G$, let $$T=g_1^{-1}ag_1g_2^{-1}ag_2\dotsm g_n^{-1}ag_n.$$ We assume that $a$ is cyclically reduced; i.e., $l(a)$ is the minimum of the set $\{l(hah^{-1}): h\in G\}$.

What can we say about $l(T)$ in terms of $l(a)$? Is it known that $l(T)\geq l(a)$? (I conjectured this, but can't prove it.)

Background:

The length $l(a^n)$ in terms of $l(a)$ is easily determined. Products of conjugates of $a$ is a natural generalization of powers of $a$. Thus it's natural to ask the relation of the lengths with respect to this generalization.

I came up with this when studying orderability of groups.

A related result of Weinbaun (1972) tells us that $T$ can't be a subword of $g$. So for example, if $g=aba^{-1}b^{-1}$ then $T$ can't be $ab$. But his result doesn't rule out the possibility that, for instance, $T=ba$.