Let $X$ be an infinite-dimensional separable Banach space and let $(x_n)_{n\in\mathbb{N}}$ be an arbitrary infinite sequence of unit vectors whose span is dense in $X$ (but they do not necessarily form a basis). Say that a linear functional $f\in X^*$ is $c_0$-like, resp. $c_{00}$-like (with respect to $(x_n)_n$) if $\lim_{n\to\infty} f(x_n)=0$, resp. for all but finitely many $n$, $f(x_n)=0$.
Is the set of $c_0$-like, resp. $c_{00}$-like functionals $w^*$-dense in $X^*$? To avoid some easy counter-examples, let's say that $\inf_{i\neq j} \|x_i-x_j\|>0$.
I have neither a proof nor a counter-example.