  We consider constant scalar curvature K\"{a}hler metrics on a smooth minimal
model of general type in a neighborhood of the canonical class, which is the
perturbation of the canonical class by a fixed K\"{a}hler metric. We show that
sequences of such metrics converge smoothly on compact subsets away from a
subvariety to the singular K\"{a}hler Einstein metric in the canonical class.
This confirms partially a conjecture of Jian-Shi-Song about the convergence
behavior of such sequences.
