  The aim of this paper is to investigate uniqueness of conic constant scalar
curvature Kaehler (cscK) metrics, when the cone angle is less than $\pi$. We
introduce a new H\"older space called $\cC^{4,\a,\b}$ to study the regularities
of this fourth order elliptic equation, and prove that any $\cC^{2,\a,\b}$
conic cscK metric is indeed of class $\cC^{4,\a,\b}$. Finally, the reductivity
is established by a careful study of the conic Lichnerowicz operator.
