Norm continuity of weakly continuous mappings into Banach spaces

Let T be the class of Banach spaces E for which every weakly continuous mapping from an α-favorable space to E is norm continuous at the points of a dense subset. We show that:
• T contains all weakly Lindelöf Banach spaces;
• l∞∉T, which brings clarity to a concern expressed by Haydon ([R. Haydon, Baire trees, bad norms and the Namioka property, Mathematika 42 (1995) 30–42], pp. 30–31) about the need of additional set-theoretical assumptions for this conclusion. Also, (l∞/c0)∉T.
• T is stable under weak homeomorphisms;
• E∈T iff every quasi-continuous mapping from a complete metric space to (E,weak) is densely norm continuous;
• E∈T iff every quasi-continuous mapping from a complete metric space to (E,weak) is weakly continuous at some point.