There is probably an embarrassingly simple counter-example to my question but I couldn't figure it out myself. Let me give it a try here.

A Banach space $X$ is Grothendieck if weak*-convergent sequences in $X^*$ converge weakly (that is, with respect to the weak topology introduced by functionals in $X^{**}$). Standard examples of such spaces include reflexive spaces and $C(K)$-spaces for $K$ Stonian.

I would like to relax this condition a bit, so my question is:

Let $X$ be a Banach space such that $X^*$ is weak-separable and let $T$ be a total subspace of $X^*$. Suppose that each sequence $(f_n)_{n=1}^\infty \subset T$ which converges weak, converges also weakly. Can we conclude that $X$ is Grothendieck?

I guess that one can take a separable, non-reflexive space $X$ as a counter-example.