  A Banach space $X$ is said to have property (K) if every $w^*$-convergent
sequence in $X^*$ admits a convex block subsequence which converges with
respect to the Mackey topology. We study the connection of this property with
strongly weakly compactly generated Banach spaces and its stability under
subspaces, quotients and $\ell^p$-sums. We extend a result of Frankiewicz and
Plebanek by proving that property (K) is preserved by $\ell^1$-sums of less
than $\mathfrak{p}$ summands. Without any cardinality restriction, we show that
property (K) is stable under $\ell^p$-sums for $1<p<\infty$.
