Twisted conjugacy in free groups and Makanin's question

We discuss the following question of G. Makanin from ``Kourovka notebook’': does there exist an algorithm to determine is for an arbitrary pair of words $U$ and $V$ of a free group $F_n$ and an arbitrary automorphism $\phi \in Aut(F_n)$theequation$\phi (X) U = V X$solvablein$F_n$? We give the affirmative answer in the case when an automorphism is virtually inner, i.e. some its non-zero power is an inner automorphism of $F_n$.