We prove that every separable Banach space containing an isomorphic copy of
can be equivalently renormed so that the new bidual norm is octahedral. This answers, in the separable case, a question in Godefroy [Metric characterization of first Baire class linear forms and octahedral norms, Studia Math. 95 (1989), 1–15]. As a direct consequence, we obtain that every dual Banach space, with a separable predual and failing to be strongly regular, can be equivalently renormed with a dual norm to satisfy the strong diameter two property.