In this paper, we describe conjugacy geodesic representatives in any dihedral Artin group G(m), m > 2, which we then use to calculate asymptotics for the conjugacy growth of G(m), and show that the conjugacy growth series of G(m) with respect to the "free product" generating set {x, y} is transcendental. We prove two additional properties of G(m) that connect to conjugacy, namely that the permutation conjugator length function is constant, and that the falsification by fellow traveler property (FFTP) holds with respect to {x, y}. These imply that the language of all conjugacy geodesics in G(m) with respect to {x, y} is regular.

The authors would like to thank Yago Antolin, Corentin Bodart, Martin Edjvet, Thomas Haettel and Fujii Michihiko for their generous advice and helpful discussions.