We continue the previous talk and focus on $C(K)$ spaces which satisfy the Grothendieck property. In the beginning we recall some notions related to sequential convergence in Banach spaces, namely weakly Cauchy sequences, and convex block subsequences (which can be understood as a generalization of the notion of a subsequence). Finally, we will use these notions to prove that consistently every non-reflexive Grothendieck Banach space contains $l_1(2^\omega)$, and thus admits $l_\infty$ as a quotient. This, together with the result that non-Grothendieck $C(K)$ spaces contain a complemented copy of $c_0$, will give us a dichotomy result about $C(K)$ spaces.