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We use relative hyperbolicity of mapping tori and Dehn fillings of relativelyhyperbolic groups to solve the conjugacy problem between certain outerautomorphisms. We reduce this problem to algorithmic problems entirelyexpressed in terms of the parabolic subgroups of the mapping tori. As animmediate application, we solve the conjugacy problem for outer automorphismsof free groups, whose polynomial part is piecewise inner. This proposes a pathtoward a full solution to the conjugacy problem for $\mathrm{Out}(F_n)$.

Reducing the conjugacy problem for relatively hyperbolic automorphisms to peripheral components