Let $(X,\|\cdot\|)$ be a Banach space which is the dual of another Banach space. The Banach-Alaoglu theorem asserts that the closed unit ball in $X$ is compact in the weak*-topology. Assume that we have another norm $\|\cdot\|_2$ on $X$ which is equivalent to the given one, so that there is $C\geq1$ with

$\forall x\in X:\quad C^{-1}\|x\|\leq\|x\|_2\leq C\|x\|$.

Is it true that the closed unit ball in this second norm is also weak*-compact?