Total subspaces with long chains of nowhere norming weak*-sequential closures

If a separable Banach space X is such that for some nonquasireflexive Banach space Y there exists a surjective stricty singular operator $T : X 'to Y$ then for every countable ordinal $'alpha$ the dual of X contains a subspace whose weak* sequential closures of orders less than $'alpha$ are nowhere norming and whose weak* sequential closure of order $'alpha+1$ coincides with $X*$