Let $T\colon X\to Y$ be a bounded linear operator. Suppose that $Z$ is a subspace of $Y^*$ such that $T^*$ is [bounded below][1] on $Z$. Denote by $\text{w*-dens}\, Z$ the minimal cardinality of a weak*-dense subset of $Z$ and define $\text{dens}\, W$ as the minimal cardinality of a norm-dense subset of $W$. My question is the following.
If $T^*$ is bounded below on $Z$ does it follow that there exists a subspace $W\subset X$ with $\text{dens}\, W = \text{w*-dens}\, Z$ on which $T$ is bounded below?
The answer is yes if $Z = P^*[W^*]$ for some idempotent operator $P\colon X\to X$ and some $W$ complemented in $X$. How about in general?
Edit. The answer is no. Take a surjection $T\colon \ell_1\to \ell_2$. Then $T$ is strictly singular but $T^*$ is not.