Suppose we have a Banach space $X$ and a net of bounded operators $(T_\gamma)$ on $X$ such that $T_\gamma^*\to S$, for some bounded operator $S$ on $X^*$, where the convergence is with respect to the dual topology by the dual pairing $(S,f):=Sf$ on $B(X^*)\times X^*$. Does $S$ belong to $\{T^*:T\in B(X^*)\}$, that is, is $S=T^*$ for some operator $T$ on $X$?
I doubt it but I have no counterexample.