  In this paper, improving a preceding work, we obtain asymptotic polybalanced
kernels associated to extremal Kaehler metrics on polarized algebraic
manifolds. As a corollary, we have a stronger asymptotic relative
Chow-polystability for extremal Kaehler polarized algebraic manifolds. Finally,
related to the Yau-Tian-Donaldson Conjecture for extremal Kaehler metrics, we
shall discuss the difference between strong relative K-stability and relative
K-stability.
