The uniqueness of Poincaré type constant scalar curvature Kähler metric

Let $D$ be a smooth divisor on a closed Kähler manifold $X$. First, we prove that Poincaré type constant scalar curvature Kähler (cscK) metric with a singularity at $D$ is unique up to a holomorphic transformation on $X$ that preserves $D$, if there are no nontrivial holomorphic vector fields on $D$. For the general case, we propose a conjecture relating the uniqueness of Poincaré type cscK metric to its asymptotic behavior near $D$. We give an affirmative answer to this conjecture for those Poincaré type cscK metrics whose asymptotic behavior is invariant under any holomorphic transformation of $X$ that preserve $D$. We also show that this conjecture can be reduced to a fixed point problem.