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If a separable Banach space $X$ is such that for some nonquasireflexiveBanach space $Y$ there exists a surjective strictly singular operator $T:X\toY$ then for every countable ordinal $\alpha $ the dual of $X$ contains asubspace whose weak$^*$ sequential closures of orders less than $\alpha $ arenot norming over any infinite-dimensional subspace of $X$ and whose weak$^*$sequential closure of order $\alpha +1$ coincides with $X^*$

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