In this paper we prove, under certain condition, when $R$ is a semiprime ring with suitable characteristic restrictions, that every nonzero $(m,n)$-Jordan triple centralizer (resp., $(m,n)$-Jordan triple derivation) is a two-sided centralizer (resp., a derivation which maps $R$ into $Z(R)$). This give partial affirmative answers to two conjectures of Vukman.