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Jul 31, 2020 at 20:52 comment added user47274 A first necessary condition is that for every $g\in F_n$, the maps $\rho(g)$ and $\rho(\phi(g))$ must be conjugate in $\mathrm{Homeo}_+(\mathbb{R})$. (clearly what you ask is the stronger condition that the conjugating element can be chosen so that it is always the same). Note that this is sufficient in the case $n=1$.
Jul 31, 2020 at 8:00 history edited YCor CC BY-SA 4.0
added tags, formatting, fixed title
Jul 31, 2020 at 6:36 history asked guest CC BY-SA 4.0