The representation image unipotency of the group $F_2$ by mapping primitive elements into unipotent matrices with small Jordan blocks

It is proved that the representation image of the free group $F_2(x, y)$ in $GL(n, C))$ is an unipotent subgroup, when $(\rho (p) - E)^5 = 0$ for all primitive elements $p$ and $(\rho(\xi) - E)^2 = 0$, $(\rho(\gamma) - E)^3 = 0$ for some associated primitive elements $\xi$ and $\gamma$ of the group $F_2$ .