  We prove, among other results, that three standard measures of weak
non-compactness coincide in preduals of JBW$^*$-triples. This result is new
even for preduals of von Neumann algebras. We further provide a
characterization of JBW$^*$-triples with strongly WCG predual and describe the
order of seminorms defining the strong$^*$ topology. As a byproduct we improve
a characterization of weakly compact subsets of a JBW$^*$-triple predual,
providing so a proof for a conjecture, open for almost eighteen years, on
weakly compact operators from a JB$^*$-triple into a complex Banach space.
