The discussion in Convergence in topologies, especially the comments of GEdgar, led me to another (converse) question concerning convergence. In the paper
G. A. Edgar, A long James space, in: Measure Theory, Oberwolfach 1979, Lectures Notes in Math. 794, Springer-Verlag (1980) p. 31-37
he points out that the weak and weak* topologies in $J(\omega_1)^*$ have different Borel $\sigma$-fields. Do they have the same convergent sequences? Heuristically, they should as $J(\omega_1)$ just "differs" with a reflexive space "by one dimension".