15.9 (2002)

An automorphism $\varphi$ of the free group $F_n$ on the free generators $x_1, x_2, \dots, x_n$ is called conjugating if $x_i^\varphi = t_i^{-1} x_{\pi(i)} t_i$, $i = 1, 2, \dots, n$, for some permutation $\pi \in S_n$ and some elements $t_i \in F_n$. The set of conjugating automorphisms fixing the product $x_1 x_2 \dots x_n$ forms the braid group $B_n$. The group $B_n$ is linear for any $n \geqslant 2$, while the group of all automorphisms $\text{Aut}\,F_n$ is not linear for $n \geqslant 3$. Is the group of all conjugating automorphisms linear for $n \geqslant 3$?