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 strictly 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 not norming over any infinite-dimensional subspace of $X$ and whose weak$^*$ sequential closure of order $\alpha +1$ coincides with $X^*$