An Asplund space with norming Markuševič basis that is not weakly compactly generated

We construct an Asplund Banach space with a norming Markuševič basis such that is not weakly compactly generated. This solves a long-standing open problem from the early nineties, originally due to Gilles Godefroy. En route to the proof, we construct a peculiar example of scattered compact space, that also solves a question due to Wiesław Kubiś and Arkady Leiderman.