Separating diameter two properties from their weak-star counterparts in spaces of Lipschitz functions

We address some open problems concerning Banach spaces of real-valued Lipschitz functions. Specifically, we prove that the diameter $2$ properties differ from their weak-star counterparts in these spaces. In particular, we establish the existence of dual Banach spaces lacking the symmetric strong diameter $2$ property but possessing its weak-star counterpart. We show that there exists an octahedral Lipschitz-free space whose bidual is not octahedral. Furthermore, we prove that the Banach space of real-valued Lipschitz functions from any infinite subset of $\ell_1$ possesses the symmetric strong diameter 2 property. These results are achieved by introducing new sufficient conditions, providing new examples and clarifying the status of known ones.