Bidual of a WSC space Let $E$ be a Banach space which is weakly sequentially complete (i.e. each weak Cauchy sequence converges weakly). Must $E^{**}$ be weakly sequentially complete either? Of course, this question is interesting only for non-reflexive spaces.

This is not true.
I reproduce Remark 3. on page 101 (in the section on Banach lattices) of Lindenstrauss-Tzafriri (in the old Springer Lecture Notes 338 edition):

Reference [80] is:
William B. Johnson, A complementary universal conjugate Banach space and its relation to the approximation problem, Israel Journal of Mathematics 13(3-4) (1972), 301–310.
