  We prove that every separable Banach space containing $\ell_1$ can be
equivalently renormed so that its bidual space is octahedral, which answers, in
the separable case, a question by Godefroy in 1989. As a direct consequence, we
obtain that every dual Banach space, with a separable predual, failing to be
strongly regular (that is, without convex combinations of slices with diameter
arbitrarily small for some closed, convex and bounded subset) can be
equivalently renormed with a dual norm to satisfy the strong diameter two
property (that is, such that every convex combination of slices in its unit
ball has diameter two).
