  We use relative hyperbolicity of mapping tori and Dehn fillings of relatively
hyperbolic groups to solve the conjugacy problem between certain outer
automorphisms. We reduce this problem to algorithmic problems entirely
expressed in terms of the parabolic subgroups of the mapping tori. As an
immediate application, we solve the conjugacy problem for outer automorphisms
of free groups, whose polynomial part is piecewise inner. This proposes a path
toward a full solution to the conjugacy problem for $\mathrm{Out}(F_n)$.
