Let $E,F$ be Banach spaces and let $A\subset K(E,F)$ be a subset of the space of compact operators from $E$ to $F$. A result by Kalton states that $A$ is weakly compact if and only if $A$ is WOT* compact (here WOT* denotes the dual weak operator topology, i.e. the topology defined by the functionals $K(E,F)\ni T\mapsto e''(T^*f')$, for $e''\in E^{**}$, $f'\in F^*$):

N. Kalton, "Spaces of compact operators", Math. Ann. 208, 267--278 (1974) .

On the other hand, the Eberlein-Smulian theorem tells us that weak compactness is equivalent to weak sequential compactness.

Question: Is WOT* compactness in $K(E,F)$ equivalent to WOT* sequential compactness? (maybe under some extra assumptions)