Recall that a Banach space $X$ has the property (K) if every $w^{*}$-convergent sequence in $X^{*}$ admits a convex block subsequence which converges with respect to the Mackey topology. The property (K) was invented by S. Kwapien to provide an alternative approach to some results of N. Kalton and A. Pelczynski on subspaces of $L_{1}[0,1]$. A. Pelczynski noted that $L_{1}(\mu)$ ($\mu$ is a finite measure) has the property (K). Further, Schur spaces, Grothendieck spaces and strongly weakly compactly generated spaces enjoy the property (K). It is known that $c_{0}$ and $C[0,1]$ fail the property (K). But it seems not easy to check that $C[0,1]$ fails the property (K).

**Question**. How to check that $C[0,1]$ fails the property (K)?